LESSONS IN ARITHMETIC.-XII. FRACTIONS (continued). 15. Multiplication of Fractions. To multiply by . This means to take four-fifths of the fraction; that is, it is the same thing as finding the value of the complex fraction of 3. Now, if be divided into five equal parts, i.e., if be divided by 5, we get; because, to divide a fraction by a whole number, we multiply the denominator by that number (Art. 5); and taking four of these fifth parts of 3-viz., four times we get as the required result. This result is plainly got by multiplying the numerators together and the denominators together of and, to form a numerator and denominator respectively. The same method would evidently apply to any other two or more fractions. Hence the following Rule for the Multiplication of Fractions. Multiply together all the numerators for a numerator, and all the denominators for a denominator. Obs. In multiplying fractions we can often simplify the operation by striking out or cancelling factors (as we are at liberty to do, Art. 6) which are common to the numerator and denominator of the fraction formed by multiplying the numera tors and denominators together. EXAMPLE.-Multiply together, equal to 2 x 5 x 6 x 55 3 x 8 x 11 x 108 55 Their product is 2 x 5 x 3 x 2x 11 x 5 3×2× 2x 2 x 11 x 108. Dividing by a whole number is finding how many times the divisor is contained in the dividend. Now, a seventh is contained in unity 7 times, and therefore a seventh is contained in X 7 times; 5 sevenths will be contained therefore in onefifth of this number of times, and therefore the quotient of by is, that is, , and the same method will be true for any other two fractions. Hence the following Rule for the Division of Fractions. Invert the divisor, and then proceed as in multiplication, i.e., multiply the numerators together for a numerator, and the denominators for a denominator. Obs. In performing the process, the Obs. of Art. 15, with reference to cancelling factors which are common to both numerator and denominator, must be attended to. 17. By this and the foregoing rules we are able to simplify complex fractions. 3. 49,7% × 681 4 22514 x 321. 2 71 x 81. 5. 10001 × 111. 6. 476851 × 376). # 39, add, and multiply the result the indicated expressions :— EXAMPLE. TO 220 71 by 8 DECIMALS. 1. FRACTIONS, the denominators of which are 10 or any power of 10, are called Decimal Fractions, or, more shortly, Decimals. Thus, are Decimal Fractions. Such fractions are represented by a method of notation which is an extension of that employed for whole numbers. In whole numbers the figures increase in a tenfold ratio from right to left; or, what is the same thing, decrease in a tenfold ratio from left to right. If we extend this method of representation towards the right beyond the units' place, any figure one place to the right of the units' place will be one-tenth of what it would be if it were in the units' place, and will thus really denote a decimal fraction; any figure two places to the right of the units' place will be one-hundredth of what its value would be if it were in the units' place; and so on for any number of figures and places. THE expression of 1 and r in composition is effected thus :--p, with 1, becomes pl;p, with r, becomes pr; |t, with 7, becomes [ ; t, with r, becomes tr. As a curve carnot receive a hook on both sides of the stroke (for such characters as could not be written both accurately and quickly), and as the r compounds are much more frequent than the compounds, a hook prefixed to a curve always adds r to the primary wl, explained par. 31), thus— Hence, if we choose some means of indicating the point in any row of figures at which the units' place occurs, we can write down any decimal fraction without the trouble of express-letter (except in the case of ing the decimal denominators. This is done by putting a dot, thr; \ ƒ, with r, becomes fr, n, with r, becomes nr; m, with r, becomes or decimal point, as it is generally called, between the figure inth, with r, becomes the units' place and the figure in the place to the right of it, which we may call the tenths' place. Thus, 14 would mean 1+ 3 would mean; 3.14159 would mean 3 +1% +YOU+ 1850 + 10000 + T00003. 2. We generally speak of any figure in a decimal as being in such a place of decimals. Thus, in the last example we should say that the 5 is in the fourth place of decimals, the 9 in the fifth place, and so on, reckoning from left to right. Observe that the denominator of the fraction corresponding to the figure in any decimal place is unity followed by the same number of ciphers as the decimal place; or, what is the same thing, that the power of 10, which is the denominator, is the same as the number of the decimal place. 3. The figures 1, 2, 3, 4, 5, 6, 7, 8, 9 in a decimal are sometimes called significant figures, or digits. Thus in such a decimal as 0002356, we should say that 2 is the first significant digit, because it is the first figure which indicates a number, the ciphers only serving to fix the place in which the 2 occurs. 4. To express a Decimal as a Vulgar Fraction. Rule for expressing a Decimal as a Vulgar Fraction. Write down the figures which compose the decimal (both integral and decimal part, if there is an integral part) for the numerator, omitting the decimal point; and for the denominator put 1, followed by as many ciphers as there are decimal places in the given decimal. 5. Conversely, if we have a fraction with any power of 10 for its denominator, we can express it as a decimal by placing a decimal point before as many right-hand figures in the numerator as there are ciphers in the denominator. Thus mr. 48. A series of curved hooked letters to represent the addition of 7, is produced by making a LARGE hook. Of this series, only f, vl, shl, nl, will be required in writing English, and these letters occur but seldom. The principle of hooking on / and r to the other letters, does not apply to the letters and themselves. 49. The most useful letters in the curved and r series, fl, vl, fr, vr, thr, have duplicate forms, namely, the opposite curves of ƒ and th in addition to the regularly formed letters; because the downward r and s do not require to be hooked for rr, sr. 50. In these hooked letters, the hook must not be considered as r, and the stem as the primary letter, but the whole form must be taken to represent the consonant diphthong pr, considered as a whole; and in no case can the r be read first; thus cannot be considered as rp, and used for reap. The left-hand hook was selected for the r series, and the right-hand hook for the 7 series, in the straight letters, because the combinations pr, tr, etc., occur five times as often as pl, tl, etc., and the left-hand hook is the best sign for writing, both when occurring singly, and when joined to other letters. 51. If the Right hand be held up, with the first finger bent, the outline of tr will be seen; and by turning the hand round to the following positions, all the double consonants of the pr series will be formed; thus 52. When speaking of these double consonants, as, for instance, in a phonographic class, it will be found convenient to pronounce them with monosyllabic names; thus, tr should not be called te, ar, but ter ; \ per; | tel; \pel, etc. A distinction is thus made between p, 1, pronounced as two letters, and pl, pronounced as one. The former would mean, the latter So the compounds in par. 17 may be named kess,tess, sek, etc. 53. These double consonants are vocalised like the single ones; thus 1. tree, pray, apply,mutter. e sp, P st, a sk. 60. Except when spr, str, skr, occur initially, the circles and the hook of the double letter must be distinctly expressed; thus express, قط Exeter, pester, register, disclose excursive, destroy, prosper. 54. Shl, shr, shn, and rt, upward, and In, Int, downward, must | right-hand side; thusnever stand ALONE, because they would then be read as other letters. 55. DOUBLE CONSONANTS.-As the stroke s hooked, thus), is not required for sr (the circle s joined to the downward r, being equally serviceable), and as the downward r, hooked for rr, would be almost useless, the two forms are given to fr, or, and their heavy strokes to vr, ar, as extra signs. These duplicate forms are distin guished, in ordinary printing, thus:-"fr, vr, Jr, dr" represent the alphabetic forms C, CC, and “fr, vr, Or, đr” the EXTRA forms). The upward letters for w, wh, y, and both the upward and downward h, may be hooked at the end for n, and halved for t or d. A letter with an initial or final hook (or both an initial and final hook) may be written half-length for the expression of either tord; thus, kit or kid, wt or wd, mnt or mnd, (down) Int or Ind, rnt or rnd, wnt or wnd. (up) 1. abridge, abroad, April, acre, across, address, agree, altar, archer. 2. baker, banner, batter, butter, beaver, beggar, brace, bridge, brick. 3. copper, cooper, crib, cram, crape, crash, crawl, cress, cruel, crow. 4. dagger, decree, dinner, diver, drab, dram, draw, drover, dreary. 5. eager, either, elder, empress, father, favour, fetter, fibre, free, froth. 6. grab, grace, grade, Greek, grapes, Hebrew, honour (o, nr), hatter. 7. increase, jabber, keeper, ladder, ledger, leper, lever, Lodger, lucre. 8. major, maker, manner, meagre, negro, neither, nipper, neighbour. 9. oppress, other, otter, over, owner, opener, offer, ochre, ogre. 10. packer, paper, phrase, potter, pray, preach, prig, prime, pucker 11. quaker, quaver, rammer, rather, reaper, rider, river, rigour, robber. 12. sceptre, shiver, shrug, shriek, shrill, silver, skipper, spider, sugar. 13. taper, tatter, thrash, three, throb, tiger, trace, trail, truck, trill. 14. upper, usher, utter, viper, vapour, wafer, wager, water, writer. Write with the hook 7 1. able, ably, addle, amble, angle, ancle, apple, apply, applause. 2. battle, bauble, beadle, black, blade, blame, blaze, blight, blush. 3. cable, cackle, clad, claim, clap, clash, clear, clime, club, clutch. 4. dabble, daggle, devil, double, dribble, eagle, eclat, edible, employ. 5. fable, fickle, fiddle, final, flame, flap, flash, flee, flight, flower. 6. gable, glare, glass, gleam, gloat, globe, gloom, glory, glow, glue. 7. hackle, hobble, idle, imply, kennel, kettle, label, ladle, legal, libel. 8. metal, muckle, muddle, needle, nettle, nibble, nipple, noble, nobly. 9. o'clock, paddle, pannel, papal, patter, people, pickle, place, platter. 10. quibble, rabble, radical, rattle, riddle, reply, rankle, ripple. 11. shuffle, shuttle, simply, stable, staple, steeple, stifle, suitable. 12. table, tackle, title, tittle, tipple, total, treble, triple, tunnel, vocal. 57. From the pr series of double consonants, a series of treble consonants is formed by making the hook into a circle; thus 61. When one of the 7 or r hooked letters follows the circle s, and a perfect hook cannot be formed, an imperfect one will suffice; thusexplore; or it may be omitted; thus, subskibe superskibe for superscribe. for subscribe, 62. After and d, the circle may be turned to the right in order o form the treble consonants skr, sgr; thus describe, 63. In the combinations dsk, dsg, the circle is written on the other side; thus, desk, Lo disguise. 64. S is joined to a consonant of the pl series thus, supply, settle, saddle, e sickle, suffer. 65. Occasionally it is more convenient to write the single consonants than to use a double letter of the pl and pr series; thus, is the best form for sensible. EXERCISE 16. Write with the spr character:-Sprain, sprat, sprawl, spray, spread, spree, sprig, sprightly, spring, sprinkle, spright, spruce. Write with the str character:-Strain, straight, strand, strange, strangle, strap, straw, stray, streak, stream, street, strength, stress, stretch, stride, strife, strike, string, strip, stripe, strive, stroke, stroll, strong, strop, struggle, strut. Write with the skr character:-Scramble, scrap, scrape, scratch, scrawl, scream, screech, screen, screw, scribble, scribe, scrip, scroll, scrub, scruple, scrutiny. GRAMMALOGUES. 66. Phonography may be written either IN FULL or IN BRIEF. Full Phonography signifies the expression of every vowel and consonant in a word by its shorthand letter. In Abbreviated Phonogra phy every word of frequent occurrence is represented by one or more of its prominent letters. These words are called GRAMMALOGUES, of LETTER-WORDS, and the letters that represent them are called LOGOGRAMS, or WORD-LETTERS. Thus, each of the following words in line 1 is represented by the under-written shorthand letter in line 2, which letter forms part of the word when written in full, as in line 3 1. of, to, for, be, are, have, which, from. 67. There are two Styles of Abbreviated Phonography; the First is used in correspondence, and for general purposes; and the Second is employed by reporters. They differ chiefly in the number of grammalogues employed. In the First or Corresponding Style about 150 are used, but the pupil will not require more than the two dozen given below, in the present stage of his progress. 68. The stroke-vowels aw, o, oo, are used as logograms in each of these directions 、. The dashes/ represent the common words and, he; these signs being serviceable in Phraseography, where the vowel dots of these words could not be used. The series of vowel grammalogues is all, too, two; oh, owe, he; awe, who; of, to; on, but; and, should. THE LEARNER'S LIST OF GRAMMALOGUES, OR 69. The following frequently occurring words are expressed by the dip 70. The words marked (*) are written above the line; the others rest upon the line. A dash (written upward) is used for and, because the dot vowel of the word is required for a, and a dash may be joined to other words in reporting. For the same reason a stroke rather than a dot is used for he. When the upward r is written by itself, it should make an angle of 60 degrees with the perpendicular; and ch, when standing alone, should make an angle of 30 degrees. That is written by the sign for th shortened, the rule for which will be explained in a future lesson. THE CIRCLE S. 71. As the reading of the circle 8, when joined to another consonant, commonly occasions some difficulty to the learner, it may be well to repeat the rule that applies to it. S at the beginning of a word is always read first, that is, before I seat, the vowel; thus, sat, sight, soon. S at the end of a word is read last; thus,face, To mouse. 72. As mentioned in the last lesson, par. 36, when it is required to place a vowel to s or z, either before or after, the stroke-letter must be used; thus 2 ask, ? G east, ), see, so; ) ease, whose, lazy. wondrous powers. PREFIX CON, AND AFFIX ING. 73. The prefix con or com is expressed by a light dot at the commencement of a word, thus ¿ contain, ġ construe, \v comply, compose. Translation. ។ x2 There was once on a time a good little dwarf named "Try," who was so powerful that he overcame every thing that he attempted; and yet was so small that people laughed when they were told of his But the tiny man was so kind at heart, and loved so much to serve those who were less able than himself, that he would go and beg of those who knew him better, to intercede for him, that he might be allowed to help them out of their troubles; and when once he had made them happy by his noble deeds, they no longer despised him, or drove him away with sneers, but loved him as their best friend. Yet the only return this good dwarf sought for all his services, was, that when they knew any his favour, and recommend them to "Try." 74. The affix ing is expressed by a light dot at the end of a word; one who wanted a helping hand, they would say a good word in thus ♣↓ eating, C. talking, ¿, taking. TRY. 1. M ་ X 1x X дх 5. 6. " The student is now prepared to write the following easy sentences in shorthand. To assist him in the selection of the shorthand letters, we have placed the correspouding phonotypes under each word, except the grammalogues. The vowel sounds that are necessary to be expressed are put in italic. It is not necessary to write every vowel in shorthand. The downward L, R, and H, are marked by small capitals. The common "1" "r," and "h," therefore signify the upstrokes for these letters. When two consonants are placed together, without a space between, as "br, pl," the hooked double letters given in the last lesson are to be used. Write the consonants first, according to the Key placed underneath each word, and then fill in the vowels. Examples. God, whom you see me daily worship, whom I daily call upon to bless both you and me and all mankind; whose wondrous acts are recorded in those Scriptures which you constantly read,-God, who created the heavens and the earth; who appointed his Son Jesus Christ to redeem mankind:-God, who has done all these great things, who has created so many millions of men, with whom the spirits of the good will live and be happy for ever;-this great God, the Creator of worlds, of angels, and of men, is your Father and Friend. which is dangerous, but the abuse of them;-it is not when they are occasionally, but when they are constantly pursued; when the love of sional indulgence, it becomes an habitual desire. It is not, therefore, the use of the innocent amusements of life amusement degenerates into a passion; and when, from being an occa In every pursuit, whatever gives strength and energy to the mind of man, experience teaches to be favourable to the interests of piety, of knowledge, and of virtue ;-in every pursuit, on the contrary, whatever enfeebles or limits the powers of the mind, the same experience ever shows to be hostile to the best interests of human nature From the first hour of existence to the last,-from the cradle of the infant, beside which the mother watches with unslumbering eye, to the grave of the aged, where the son pours his bitterest tears upon the bier of his father,-in all that intermediate time, every day calls for exertion and activity, and moral honours can only be won by the steadfast magnanimity of pious duty. They say they have bought it.-Bought it! Yes;-of whom ?-Of the poor trembling natives, who knew that refusal would be vain; and who strove to make a merit of necessity, by seeming to yield with grace, what they knew they had not the power to retain. It is not the lifeless mass of matter, he will then feel, that he is examining, it is the mighty machine of Eternal Wisdom : the workmanship of Him, in whom everything lives, and moves, and has its being. When suffering the inconveniences of the ruder parts of the year, we may be tempted to wonder why this rotation is necessary;-why we could not be constantly gratified with vernal bloom and fragrance, or summer beauty and profusion. Then a spirit passed before my face; the hair of my flesh stood up: Men are more what they are used to, than what they are born to, it stood still, but I could not discern the form thereof: an image was mor wo t de dan wot de men ust born The lessons of life are lost, if they do not impress us with the Lesns lif lost if de du not impres ns before mine eyes:-There was silence, and I heard a voice-Shall mortal man be more just than God? 58. The dash is sometimes to be read as a note of interrogation. Examples. Is it not enough to see our friends die, and part with them for the remainder of our days-to reflect that we shall hear their voices no more, and that they will never look on us again-to see that turning to corruption, which was but just now alive, and eloquent, and beautiful with all the sensations of the soul? He hears the ravens cry; and shall he not hear, and will he not avenge, the wrongs that his nobler animals suffer-wrongs that cry out necessity of making ample allowance for the immature opinions against man from youth to age, in the city and in the field, by the way nessiti mekn (dot) ampl als aus of others. pârs immatur opinions READING AND ELOCUTION.-VI. PUNCTUATION (continued). IX. THE DASH (continued). 56. THE dash sometimes precedes something unexpected; as when a sentence beginning seriously ends humorously. Examples. Good people all, with one accord, lament for Madam Blaize : who never wanted a good word-from those who spoke her praise. The needy seldom passed her door, and always found her kiad; she freely lent to all the poor-who left a pledge behind. She strove the neighbourhood to please, with manner wondrous winning; and never followed wicked ways-except when she was sinning. At church, in silks and satin new, with hoop of monstrous size; she never slumbered in her pew-but when she shut her eyes. Her love was sought, I do aver, by twenty beaux, and more; the king himself has followed her-when she has walked before. But now her wealth and finery fled, her hangers-on cut short all; her doctors found, when she was dead-her last disorder mortal. Let us lament, in sorrow sore; for Kent Street well may say, that had she lived a twelvemonth more-she had not died to-day. 57. The dash is sometimes used with other pauses to lengthen them. and by the fireside? Can we view their bloody edicts against us-their hanging, heading, hounding, and hunting down an ancient and honourable name-as deserving better treatment than that which enemies give to enemies ? Are these the pompous tidings ye proclaim, lights of the world, and demi-gods of fame ? Is this your triumph-this your proud applause, children of truth, and champions of her cause? Was there ever a bolder captain of a more valiant band? Was there ever-but I scorn to boast. And what if thou shalt fall unnoticed by the living-and no friend take note of thy departure? Seest thou yon lonely cottage in the grove-with little garden neatly planned before-its roof deep-shaded by the elms above, mossgrown, and decked with velvet verdure o'er? What shall we call them ?-piles of crystal light-a glorious company lighting systems with their joyous beams. of golden streams-lamps of celestial ether burning bright-suns 59. The dash is sometimes to be read like a note of exclamation. Examples. What dreadful pleasure! there to stand sublime, like shipwrecked mariner on desert coast, and see the enormous waste of vapour, tossed in billows lengthening to the horizon round, now scooped in gulfs, with mountains now embossed-and hear the voice of mirth and song rebound, flocks, herds, and waterfalls, along the hoar profound! The chain of being is complete in me; in me is matter's last gra-lation lost, and the next step is spirit-Deity! I can command the lightning, and am dust! Above me are the Alps, the palaces of Nature, whose vast walls have |