Answers.-(1.) 1, (2.) 1; (3.) &; (4.) 1; (5.) 19; and Prob. 4; (6.) 102.

Answers,-(1.) 54; (2.) 3953; (3.) 1733; (4.) 4012; (5.) But 42-27 50; and (6.) 26891.

RULE 1.-When the given fractions have a common denomi- In working this example, as above, we have first to take 1 from 7 nator, subtract the less numerator from the greater, and under it which leaves 6; then 18; next, 48-13-18; and 8+ place the common denominator, for the difference or remainder; whence 6 is the answer required. required. The difference should be reduced to its lowest terms, and if it be an improper fraction, it may be reduced to a whole or mixed number, by Problem III.

RULE 1.-When the given fractions to be multiplied are proper fractions, multiply all their numerators together for the numerator of the product, and all their denominators for the denominator of the product; the result placed in a fractional form will be the product required.

The reason of this rule may be thus explained; take, for example, two proper fractions and, and let it be required to find their product. You will recollect that the fraction may be considered as of 5 (see No. 7, p. 133, vol. ii.); you will also remember that to multiply the fraction by 5, we must multiply the numerator only, and that this product will be 5 (see No. 11, Principle 2, p. 134); now we wish to have the product of 3 by ☀ of 5; we must therefore take of the product of this fraction by 5, that is, we must take of 12; but, by No. 8, Def. 3, p. 133, this becomes, the proper answer to the question. The process, then, is plainly this; in order to multiply one fraction by another, you first multiply the multiplicand by the numerator of the multiplier, and then divide the product by its denominator (see No. 12, Principle 3, p. 134); but this process is equivalent to that expressed in the words of the rule; for multiplying the numerator of a fraction by a whole number increases its value, and multiplying the denominator by the same diminishes its value; and the numerator and denominator of the multiplier are considered as whole numbers when taken by themselves.

Another view of the reason of this rule may be taken perhaps with more simplicity and propriety as follows: When we multiply a fraction by any number we repeat or take it as many times as that number denotes; when we multiply it by unity or one we take it one time; and when we multiply it by a proper fraction we take it a part of a time; therefore, in the latter case, we must take such a part of the multiplicand as that part denotes; but to take such a part of a fraction as another fraction denotes is to take a fraction of a fraction, which, according to No. 8, Def. 3, p. 133, is to form a com o nd fraction; and this requires the operation described in the rule; see the place to which we have just referred.

From the preceding observations we draw the following important observations, viz.: That whatever be the nature of the multiplicand, 1, if it be multiplied by a whole number greater than unity, it is increased; 2, if it be multiplied by unity itself it is neither increased nor diminished; 3, and, if it be multiplied by a proper fraction it is diminished, that is, the product is always less than the multiplicand.

When the product of several fractions is required, the preceding rule is applicable to any two of them; it is then applicable to the product of these two, and another of them; then to the product of these three and another of them; and so on, whatever be their number; so that the rule for reducing compound fractions to simple ones, and for multiplying any number of fractions together is the same. This rule is, therefore, capable of the same abridg. ment, by the method of cancelling factors, explained at No. 15, p. 328, col. 2, where the rule for this process is given.

N. B.-The answer obtained by the rule above must be reduced to its lowest terms.

EXAMPLE.-Multiply

together the following (1.) § × 1; (2.) îî×3× }; and (3.) ! × × Here, the operations are indicated as follows:

factors.

Or, X X

strikingly useful; for, in (1.) the factors common to both nume-
In these examples, the process of cancelling the factors appears
rator and denominator are 4 and 3, which being left out, gives the
result, without the trouble of reducing the product to its lowest
terms; in 2.) the common factors are 2, 5, 3 and 11, and the
result; in (3.) the common factors are 2, 3, 4, 5, and 6, and
the result is 4.
EXERCISES.

Find the products of the following fractions :

towers A M, B N, and CP by a, b, and e, respectively. Let z denote fractions :the length of the ladder, which is either M O, NO, or P o, if we sup×X.pose o to be the point where the ladder must be placed, in order to reach the tops of the towers, at M, N, and P, respectively. It is plain that as the heights of the towers are of different lengths, if we suppose the lengths a, b, and e to be written in the order of their magnitude, a being the smallest, and c the tallest, that the by cancelling point o will be nearest to c and nearer to в than to A. From the point A, draw AT perpendicular to BC; from the point o, draw os perpendicular to BC, and OR to AT; then ROST is a right angied parallelogram, and A Ro a right angled triangle. Since by hypothesis, the angles MA O, N BO, and P Co are right angles, we have, by the 47th of Euclid's Book I., the following equations:(1.) ▲ o2 = x2—a2, (2.) в o2= x2 — b2, and (3.) co2= x2-e2, and denoting Ts by z, and os or RT by y, we have also, BS+SO2 Bo, and cs+so co2, or, by substitution, from the preceding, (4.) {1d+32 + y2= x2-62, and (5.) {{d—z}2+ y2= x2—c2; whence, by subtracting equation (5.) from equation (4.),

660 2 6930 21'

4 X 5 X 33

9 70

11 × 9 × 70

1

by the rule.

2 by cancel.
factors.

21'

יך

by the rule.

On se plaint de l'issue de tel événement: La fortune a trihi nos e, efforts, dit-on; c'est dire en d'autres termes: Il est arrivé un résultat sans cause. Pourquoi ces plaintes d'enfant ? Ce qui est arrivé devait arriver. Votre maison s'est écroulée; c'est qu'elle était mal étayée. Le peuple a couvert d'acclamations ses oppresseurs: c'est parce que le peuple n'est pas assez avancé pour comprendre ses véritables intérêts. La fortune n'a rien à faire là-dedans: au lieu convient à des créatures raisonnables.-J.-B. Say.

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LESSONS IN PHONETIC SHORT-HAND.-No. VI.

By ALEX. MELVILLE BELL, F.R.S.S.A.,

Professor of Elocution and Vocal Physiology, Member of the British Phonetic Council, Author of the "Principles of Speech and Elocution "The Elocutionary Manual "-" Steno-phonography,"-&c.

68. THE subordinate words given in the preceding lesson were uncontracted, except in size; written with all their elements, only without full-sized characters. The following list contains such words as are represented by a simplification of their alphabetic form, as ƒt, for after; r ring, for round; bh, for behind; dr, for under, &c.

69. The letters printed in italics in the list are those used to denote the words. Illustrations of the different forms are given, though, as they are all merely alphabetic, the student should be able to supply them for himself from the descriptive printing; and he is recommended to write them from this,-covering down the marks, -as the readiest means of fixing them upon the memory.

indicates

once, after all, quite so, quite as, quite enough, quite alone, quite another, except that, except as, except among, except indeed, except only, except some, except this, except which, except such, little less, little (or) none, large enough, as large as, perhaps so, perhaps always, perhaps indeed, accordingly therefore, rather than, rather far, rather near, rather better, rather less, rather soon, rather that, rather large, rather much, hardly (so) much, such as, as such, as much as, as little as, thenceforward, thitherward, every one, very soon, very far, altogether, together with, whichever, her own, by and bye, very little, very well.

II. Write the following words, by adding the letters in italics to the subordinate words, omitting the letters within brackets:

One thing), someth (ing), same thing), someth(ing) else, anything), justly, larger, largest, largely, likewise, underneath, somet (ime), somet(ime)s*, same t(ime), often?(ime)s*,

[As in the preceding lesson, an asterisk (*) after s, the hooks; after r, the horizontal r; after m, n, or ng, a ring: far (the)r*, far(the)st, never(the) less, here (to) fore. () signifies write upwards; and (1), write backwards].

72. Contracted letters never having vowels before them in full alphabetic writing, the addition of a dot before any small character, simple or compound, will furnish a distinctive symbol for any word to which it may be appropriated. This principle is employed to distinguish the past from the present form of the auxiliary verbs do, can, may, shall, and will. Thus. A dot before do (d)

is

can (k) is
may (m')

did.

......

could.

is

might.

shall (shl) is

should.

will (wl).

is

would.

t (to, it)

w (we)

'k (like)

is

r, ring (round) is