sense. Good writers endeavour to avoid requiring words fall from her mouth. Coleridge somewhere either parenthetic marks or dashes, both of which indi- tells that he was once much prepossessed in favour of cate irregularities of thought and expression. an individual whom he met at a dinner-table, and who appeared a dignified and respectable person, until, CONCLUSION. some kind of fruit being introduced, he heard him exWe have now explained the Etymology and Syntax claim, ' Oh, them's the jockies for me!' Words are of the English tongue, as far as our limits permit; the exponents of conditions of mind, and when mean and in drawing to a close, we may be allowed to im- ones are used, we unavoidably suppose the condition of press on our readers the value of the science which we mind to be mean. have been endeavouring to expound. If they have intelligently gone along with us in our various remarks, ERRORS IN PRONUNCIATION, they will not be surprised when we assert that this de- The interchange of w for v, and v for w, and the putpartment of human knowledge, if skilfully cultivated, ting of the sound of h before words where it is inapprowill be productive of very valuable results. To under- priate, and taking it away where it ought to be. Exstand the grammar of a sentence, is nothing more or amples-Vill you vait to get some vine and wictuals ? less than to understand its sense, and to see clearly An ’ard-boiled hegg. how its various parts are connected; while in learning The sound k instead of g at the ends of words, to recognise the different modifications that words un- Examples-Somethink, nothink. dergo, and the different arrangements of which they The addition of r at the ends of words ending in are susceptible, to express difference of thought, we vowels. Examples– Idear, windor, Elizar. have exercised many of the mental faculties, and in so Changing the termination en, ain, or eign, into ing ; far laid the foundation of what is much wanted-a just as garding for garden, founting for fountain, sovering system of Logic. for sovereign, and the like. The sources whence the student will derive effectual aid in the prosecution of this interesting subject, we UNGRAMMATICAL FORMS. have already pointed out incidentally ; but let no one Between you and I, there is a great want of conlament too much though he should not have access to scientiousness in most partisans. Correction-Between them. Rather let him, by additional thought on his you and me, &c. own part, make up for the deficiency, and he may rest I am not so proud as him. Cor. As he. assured that, by accustoming himself to mark the diffe- You will do it better than her. Cor.-Than she. rent modes of expression he meets with in reputable May thou as well as me be meek, patient, and forauthors, a system of grammar will evolve itself, which giving. Cor.--As well as 1, &c. will be all the more valued—if we may not say valuable While the house was being built. Cor.-While the - that it has been wrought out by his own exertions, house was in the course of being built. and not received by tradition or passively from the He don't go to town to-day. Cor.-He does not go hands of another. Following this plan, the real method to town to-day. of induction, he will either reproduce the rules which I rather think he is out of town. Cor.-I believe he we have set before him, or else see their erroneousness. is out of town. So that, in either case, we shall deserve well of him; I had better go myself. Cor.-It were better that I for, if we are right in anything, we shall have served should go myself. as a guide to him; and in those points where we have I had oblige to go. Cor.--I was obliged to go. erred, we shall have put him on the way to find out John is tall in comparison to James. Cor.-John is our errors. We know very well that the pupil cannot tall in comparison with James. see with our eyes, and we have therefore only endea- He is a very rising man. Cor. He is rising very voured to direct his attention to such objects as he rapidly. may see with his own. So far as he sees, he should She readied a dish for us. Cor.-She cooked, or prebelieve, and no farther. To dogmatise is the method pared, a dish for us. of a grammatist, but our ambition has been to act the She was a superior woman, or, She was a most superior part of a philosophical grammarian, and, as such, we Cor.--Superior can only be used with regard cannot conclude without warning our readers never to to something else which is at the same time expressed : forget that words in themselves are nothing, and that thus, She was a woman much superior to the generality they are only valuable in so far as they are the symbols of her sex. of ideas. Beautifully and justly has Johnson said, This is better nor that. Cor. This is better than • Words are the daughters of earth, and things only that. are the sous of heaven.' Language is but a vehicle of Short-lived, long-lived. Cor.-Short - lifed, longthought, or, at best, its instrument, and to view it as an lifed. end unto itself,' is the vain humour of a pedant. Let The then Earl of Winchelsea; the then Mrs Bennet. none be so taken up with words as to forget solid Cor.—The Earl of Winchelsea of that time; the Mrs things. Bennet then living. He lays asleep in the cabin. Cor.—He lies asleep in the cabin. COMMON ERRORS CORRECTED. His health was drank. Cor.-His health was drunk. The remaining space of the present sheet could The dinner was all eat up. Cor.— The dinner was scarcely, we think, be better employed than in enume- all eaten up. rating some examples of the most common errors in I went to table and eat very heartily. Cor.-I went the pronunciation and selection of words. In every to table and ate very heartily. part of the country there are some peculiar vices A couple of shillings. Cor.-Couple can only be proof speech, which have been handed down from one perly applied to objects in connection; as, a married generation to another, and are generally so inveterate couple, a couple of pointers. in most minds, from the effect of early habit, that no John, Jaines, and Robert, were sober workmen, the cultivation which the mind may receive in mature latter particularly so. Cor.—The last particularly so life altogether obliterates them. For any one who has (the objects enumerated being more than two). occasion to mix in refined society to be thus liable Ask at him. Cor.--Ask him. every moment to the use of some barbarism of speech, The Manchester Guardian’is a well-advertised paper is a misfortune of some magnitude; for nothing tends ---meaning a paper which usually contains many adverso much to convey a mean impression of his education tisements. Cor.-- The ‘Manchester Guardian usually and habits of life. The most beautiful young female, contains many advertisements, or-enjoys a large share who, silent, appears a kind of divinity, is reduced at of the patronage of advertisers. once to common earth when we hear a few inelegant I could not give himn credit, without he changes his woman. behaviour. Cor.-I could not give him credit, unless lle gave her a beautiful book as a present: (or better) he changes his behaviour. He presented her with, or made her a present of, a I will go, except I should be ill. Cor.--Unless I beautiful book. should be ill. No less than two hundred scholars hare been edu. I saw them all, unless two or three. Cor.-I saw cated in that school. Cor.- No fewer, &c. them all, except two or three. There was a quantity of people present. Cor.— There I took some cream into a bowl. Cor.- I took some was a number of people present. cream in a bowl. It is above a year since the time that I left school. I am going for to do it. Cor.-I am going to do it. Cor.-It is inore than a year since I left school. He was a devoted antiquarian all his days. Cor.- He felt the peculiarness of his situation. Cor.—He He was a devoted antiquary all his days. (Antiquarian felt the peculiarity of his situation. In like manner is the adjective.) delicacy should be preferred to delicateness, incapability James is going to be a medical man. Cor.–James to incapableness, &c. is going to be a physician, surgeon, or medical prac- He was guilty of such atrocious conduct, that he was titioner. deserted by his friends for good and all. Cor. He was He is oftener wrong than right. Cor.—He is more guilty of conduct so atrocious, that he was entirely frequently wrong than right. deserted by his friends. I have no right to pay this tax. I have no right to be distressed by that man's conduct. Cor.--I am under OBSOLETE, AWKWARD, AND MEAN FORMs. no obligation to pay this tax. I am not obliged to I had as lief do it myself as persuade another to do suffer from that man's conduct. it. Cor.--I would as readily, &c. You will be necessitated to submit. Cor.-You will He convinced his opponent by sheer dint of argument. be obliged to submit. Cor.-Entirely by force of argument. He is not intending to purchase it. Cor.-He does Ile is not a whit better than those whom he so libenot intend to purchase it. rally condemns. Cor.---He is not in any degree, &c. Don't talk of those sort of things to me. Cor.-Don't ile stands upon the bond, and will not abate a jot of talk of that sort of things to me. Sort of things is a his claim. Cor.—He insists on the strict terms of the mean and objectionable expression. "Things of that bond, and will not in the least abate his claim. kind' is more elegant, as well as correct. Good satin, I take it, is considerably superior to The castle is seated by the Garonne. Cor.-The common silk. Cor.--I presume, &c. castle is seated beside the Garonne. You have no call to do it. Cor.--You have no oc. Lord Byron was born at London. There have been casion to do it. destructive fires at Edinburgh. Cor.-Lord Byron was I have no right to pay. Cor.-I am not bound to pay. born in London. There have been destructive fires in Politics too often sets men by the cars. When they Edinburgh. (At is only proper with respect to a small come to words, and fall out, reason is generally lost town.) sight of. I should not wonder but on this occasion I met him on the street. Cor.--I met him in the there might be broken heuds going. Cor.-- Politics too street, often cause quarrels. When men enter into controI don't know, but I will inquire at my friend. Cor. versy, and differ violently, reason is generally lost sight --Of my friend. of. I should not wonder but on this occasion they I was calling for you yesterday. Cor.-On you yes- might commit some violence on each other. terday. We shall have a regulur break-up in the ministry. Oh, I will fall, and nobody shall help me. Cor.-Oh, Cor.---We shall have a dissolution of the ministry. I shall fall, and nobody will help me. He was very dexterous in smelling out the designs of I have been to London, and am now going for Liver- his neighbours. Cor.--In penetrating, &c. pool. Cor.- I have been in London, and am now going He is a thorough-paced knave. Cor.--He is a great to Liverpool. knave. He was married on Miss Edmonstone. Cor.--He was Heretofore Hannibal had carried all before him; married to Miss Edmonstone. wherefore he had become very proud, listening to no ad. They were some distance from home when the acci- vice whatsoever; whereas Scipio invariably took counsel dent happened. Cor.--At some distance, &c. from the most sagacious of his officers.---The words in He lives opposite the Royal Exchange. Cor.-Op- Italics are all obsolete and objectionable. posite to, &c. He vist not what to do. Cor.—He knew not what to Pray, sit into the fire. Cor.- Pray, sit near the fire. do. The performance was approved of by all who under- He little wots of the storm that is brcwing. Cor.stood it. Cor.—The performance was approved by all. He is not aware, &c. They attacked Northumberland's house, whom they Topsy-turry, pell-mell, hurly-burly, having a month's put to death. Cor.- They attacked the house of North- mind for a thing, currying furour with a person, dancumberland (or the Duke of Northumberland), whom ing attendance on customers, get into a scrape, come to they put to death. the scratch, fare up, fork out, walk into him, kick up a It is true what he says, but it is not applicable to the row, raise a rumpus, and the like-All objectionable point. Cor.--What he says is true, &c. from their meanness. Together with the national debt, the greatest national We are at one on the slave question, advantages are also transmitted to succeeding genera- I happen to have a little leisure upon my hands. tions. Cor.--Also is superfluous. He might have perceived it with half an eye. Failing in his effort, he again repeated it. Cor.- My father left this morning by the mail. Cor.-- My Aguin is superfluous. father went away this morning, &c. When are you He is noway thy inferior, and in this instance is no- to leave?' is in like manner vicious. The place or ways to blame. Cor.--He is in nowise thy inferior, thing left should always be stated. and in this instance is not at all to blame. Slang phrases of all kinds should be received warily. It is neither more nor less than medicine in disguise. The least objectionable are those which merely suggest Cor.-It is simply medicine in disguise. comical ideas; those which tend to present light and · The master never challenged him for stealing. Cor. jocular views of moral error are particularly detestable. —The master never reproved him for stealing. It will be the aim of a well-bred and judicious person He charged me with want of resolution, in which he to make his discourse neither too nice and formal, nor was greatly mistaken. Cor.--He charged me with want too loose and homely, but, as far as possible, to preserve of resolution, but in this censure he was greatly inis- a medium between the select language employed in taken. literature, and the familiar, and perhaps temporary, He gave her a beautiful book in a present. Cor.- phraseology which prevails in ordinary society. 592 ARITHMETIC-ALGEBRA. 9 In the present and succeeding sheet, an attempt is made inety being expressed by a figure formed on purpose, to convey to the comparatively unlearned mind some and resembling the Arabic 5 inverted. The remaining knowledge of Mathematical science, both as regards seven letters expressed respectively 200, 300, 400, 500, ineasurement by numbers (ARITHMETIC) and measure- 600, 700, 800; and for 900 there was another inverted ment of dimensions (GEOMETRY). The sketch we offer figure. Larger numbers were represented by letters of each is necessarily brief and imperfect; but our end accented in various ways. will be gained if we afford that amount of information The Romans, from an early period, had a method on the subject which is generally possessed by persons of expressing numbers, which seems to have been at of moderately well-cultivated intellect. first independent of the alphabet. The following intel ligible account of it has been given by Professor PlayA recognition of the value of numbers is coeval with fair :—' To denote one, a simple upright stroke was the dawn of mental cultivation in every community; assumed '; and the repetition of this expressed two, but considerable progress must be inade before methods three, &c. Two cross strokes X marked the next of reckoning are reduced to a regular system, and a no- step in the scale of numeration, or ten; and that tation adopted to express large or complex quantities. symbol was repeated to signify twenty, thirty, &c. An inability to reckon beyond a few numbers is always Three strokes, or an open square [, were employed a proof of mental obscurity; and in this state various to denote the hundred, or the third stage of numerasavage nations have been discovered by travellers. tion; and four interwoven strokes M, sometimes inSome are found to be able to count as far as five, the curved M, or even divided CIɔ, expressed a thousand. digits of the hand most likely familiarising them with Such are all the characters absolutely required in a that number ; but any further quantity is either said very limited system of numeration. The necessary to consist of so many fives, or is expressed by the more repetition of them, however, as often occasionally as convenient phrase, ' a great many.' Among the North nine times, was soon found to be tedious and perplexAmerican Indians, any great number which the mind ing. Reduced or curtailed marks were therefore emis incapable of distinctly recognising and naming is ployed to express the intermediate multiples of five; figuratively described by comparing it to the leaves of and this improvement must have taken place at a very the forest; and in the same manner the untutored early period. Thus five itself was denoted by the upper Negro of Africa would define any quantity of vast half V, and sometimes the under half 1, of the chaamount by pointing to a handful of sand of the desert. racter X for ten; L, or the half of C, the mark for On the first advance of any early people towards a hundred, came to represent fifty; and the incurved civilisation, it would be found impossible to give a symbol M, or Cly, for a thousand, was split into lɔ, separate name to each separate number which they had to express five hundred. occasion to describe. It would therefore be necessary These important contractions having been adopted, to consider large numbers as only multiplications of another convenient abbreviation was introduced." To certain smaller ones, and to name them accordingly. avoid the frequent repetition of a mark, it was prefixed This is no doubt what gave rise to classes of numbers, to the principal character, and denoted the effect by which are different in different countries. For instance, counting backwards. Thus instead of four strokes, it the Chinese count by twos; the ancient Mexicans seemed preferable to write IV ; for eight and nine reckoned by fours. Some counted by fives, a number the symbols were !IX and IX'; and ninety was exwhich the fingers would always be ready to suggest. pressed by XC: This mode of reckoning by the defect The Hebrews, from an early period, reckoned by tens, was peculiar to the Romans, and has evidently affected which would also be an obvious mode, from the the composition of their numerical terms. Instead of number of the fingers of the two hands, as well as of octodecem [eight and ten--for eighteen], and novemthe toes of the two feet. The Greeks adopted this decem (nine and ten—for nineteen), it was held more method; from the Greeks it came to the Romans, and elegant, in the Latin language, to use undeviginti [one by them was spread over a large part of the world. from twenty), and duodeviginti (two from twenty). But the alphabetic characters now lent their aid to numeration. The uniform broad strokes were dismissed, Notation is the method of expressing numbers by and those letters which most resembled the several means of certain signs or figures. The representation combinations were adopted in their place. The marks of pumbers by written signs is an art generally be for one, five, ten, and fifty, were respectively supplied lieved to have taken its rise after the formation of by the letters I, V, X, and L. The symbol for a hunalphabets. One of the earliest sets of written signs of dred was aptly denoted by C, which had originally a numbers of which we have any notice, is certainly the square shape, and happened, besides, to be the initial series of letters of the Hebrew alphabet which was used of the very word centum. The letter D was very geby that people--Aleph, beth, gimel, daleth, he, vau, zain, nerally assumed as a near approximation to the symbol cheth, teth, standing respectively for the numbers one, for five hundred ; and M not only represented the two, three, four, five, six, seven, eight, nine. The Greeks angular character for a thousand, but was likewise, directly adopted this plan from the Hebrews, forming though perhaps accidentally, the first letter of the word their numbers thus :- 1 alpha, 2 beta, 3 gamma, 4 mille-Edin. Rev. No. xviii. p. 193. delta, 5 epsilon--here, having no letter corresponding The Hebrew, improved Grecian, and Roman numewith the Hebrew rau, they put in the words stisnuor Been rals were perhaps sufficient to express any single to denote six ; after which they proceeded with 7 zeta, number with tolerable precision; but it is easy to see 8 eta, &c. Before adopting this plan, they had indi- that they must have been nearly unfitted for use in cated one by iota, probably because it was the the processes of arithmetic. The Greeks certainly consmallest of their letters; five by II (P), being the first trived to overcome many obstacles in the business of letter of pente, five; ten by A (D), being the initial calculation, and even could express fractions—though, of deka, ten. After having for some time adopted the from a practice of adding from left to right, and ignoHebrew mode, they divided their alphabet into three rance of the plan of carrying tens to the higher places, classes : the first ten letters expressing the numbers their problems were at all times awkward and complifrom one to ten; while twenty, thirty, forty, and so on cated. The Romans, however, careless of old inconup to a hundred, were signified by the next nine, ' veniences, were still more awkwardly situated than No. 88. 593 NOTATION, NUMERATION. the Greeks. Let any reader just suppose, for instance, It would be impossible to calculate, even by their even so simple a question as the amount of XLVIII own transcendent powers, the service which the Arabic added to XXXIV! It is evident that placing the numerals have rendered to mankind. figures below each other, as we do with the Arabic numerals, would serve little to facilitate such a calculation. In fact, the Romans were obliged, where Numeration is the art of numbering—that is, of ex. mental calculation would not serve, to resort to a pressing any number in words. The Arabic numerical mechanical process for performing problems in arith- signs now generally in use take the following well. metic. A box of pebbles called loculus, and a board known forms:-1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The first called abacus, constituted their means of calculation; nine of these, called digits or digital numbers, repreand of these every schoolboy, and many other persons, sent, each, one of the numbers between one and nine, possessed a set. The word calculation claims no higher and when thus employed to represent single numbers, descent than from calculus, a stone or pebble. The they are considered as units. The last (0), called a board was divided from the right to the left hand by nought, nothing, or cipher, is, in reality, taken by itself, upright columns, on which the pebbles were placed, to expressive of an absence of number, or nothing; but, in denote units, tens, hundreds, thousands, &c. The labour connection with other numbers, it becomes expressive of counting and arranging the pebbles was afterwards of number in a very remarkable manner. sensibly abridged by drawing across the board a hori. The valuable peculiarity of the Arabic notation is the zontal line, above which each single pebble had the enlargement and variety of values which can be given power of five. In the progress of luxury, tali, or dies to the figures by associating them. The number ten is made of ivory, were used instead of pebbles; and after- expressed by the 1 and 0 put together-thus 10; and wards the whole system was made more convenient by all the numbers from this up to a hundred can be exsubstituting beads strung on parallel threads, or pegs pressed in like manner by the association of two figures stuck along grooves; methods of calculation still used --thus, twenty, 20; thirty, 30; eighty-five, 85; ninetyin Russia and China, and found convenient in certain nine, 99. These are called decimal numbers, from decem, departments of Roman Catholic devotion, and in seve- Latin for ten. The numbers between a hundred and ral familiar games in more civilised countries. With nine hundred and ninety-nine inclusive, are in like mansuch instruments, problems in addition and subtraction ner expressed by three figures—thus, a hundred, 100; would not be very difficult; but those in multiplication five hundred, 500; eight hundred and eighty-five, 885; and division, not to speak of the more compound rules, nine hundred and ninety-nine, 999. Four figures express must have been extremely tedious and irksome. So dis- thousands; five, tens of thousands; six, hundreds of agreeable, indeed, was the whole labour, that the Romans thousands; seven, millions; and so forth. Each figure, generally left it to slaves and professional calculators. in short, put to the left hand of another, or of several The numerals now in use, with the mode of causing others, multiplies that one or more numbers by ten. Or them by peculiar situation to express any number, and if to any set of figures a nought (0) be added towards the whereby the processes of arithmetic have been ren- right hand, that addition multiplies the number by ten; dered so highly convenient, have heretofore been sup- thus 999, with 0 added, becomes 9990, nine thousand posed to be of Indian origin, transmitted through the nine hundred and ninety. Thus it will be seen that, in Persians to the Arabs, and by them introduced into notation, the rank or place of any figure in a number Europe in the tenth century, when the Moors invaded is what determines the value which it bears. The figure and became masters of Spain. Such, in reality, ap- third from the right hand is always one of the hundreds; pears to have been in a great measure the true his- that which stands seventh always expresses millions; tory of the transmission of these numerals; but as it and so on. And whenever a has been lately found that the ancient hieroglyphical 1 2 3 4 5 6 7 8 new figure is added towards inscriptions of Egypt contain several of them, learned the right, each of the former men are now agreed that they originated in that early set is made to express ten seat of knowledge, between which and India there exist times its former value, A more points of resemblance, and more traces of inter large number is thus excourse, than is generally supposed. In the eleventh pressed in the Arabic numecentury, Gerbert, a Benedictine monk of Fleury, and rals, every set of three from who afterwards ascended the papal throne under the the right to the left hand designation of Sylvester II., travelled into Spain, and being separated by a comma studied for several years the sciences there cultivated for the sake of distinctness, by the Moors. Among other acquisitions, he gained The above number is therefore one thousand two from that singular people a knowledge of what are hundred and thirty-four millions, five hundred and now called the Arabic numerals, and of the mode of sixty-seven thousands, eight hundred and ninety. arithmetic founded on them, which he forthwith dis- Higher numbers are expressed differently in France closed to the Christian world, by whom at first his and England. In the former country, the tenth figure learning caused him to be accused of an alliance with expresses billions, from which there is an advance to evil spirits. The knowledge of this new arithmetic was tens of billions, hundreds of billions, trillions, &c. about the same time extended, in consequence of the In our country, the eleventh figure expresses ten intercourse which the Crusaders opened between Eu- thousands of millions, the next hundreds of thourope and the East. For a long time, however, it made sands of millions, the next billions, &c. The two a very slow and obscure progress. The characters methods will be clearly apprehended from the follow. themselves appear to have been long considered in ing arrangement:Europe as dark and mysterious. Deriving their whole efficacy from the use made of the cipher, so called Units. from the Arabic word tsaphara, denoting empty or void, this term came afterwards to express, in general, any Thousands. secret mark. Hence in more troublous times than the Tens of thousands. Tens of thousands. present, a mode of writing was practised, by means of Hundreds of thousands Hundreds of thousands. marks previously concerted, and called writing in cipher. The Arabic characters occur in some arithmetical tracts Hundreds of millions. composed in England during the thirteenth and four Thousands of millions. teenth centuries, particularly in a work by John of Tens of billions. Halifax, or De Sacrobosco; but another century elapsed Hundreds of thousands of mil. Hundreds of billions. before they were generally adopted. They do not ap lions. pear to have settled into their present forms till about Tens of billions. Hundrodis of trillions, &c. the time of the invention of printing. Hundreds of billions. Millions. Units. Thousands. Tens of thousands. ENGLISH, Tens. FRENCH Millions. Millions. Billions. Ten thousands of millions. Billions. Trillions. SIMPLE OR ABSTRACT NUMBERS. 5 For practice in Notation and Numeration, the reader This table is so well known, that it is almost supershould write down large numbers alternately in words fuous to explain that, when any number in the top and figures; at first assisting himself by the use of row is multiplied by any number in the left-hand side commas, but gradually dispensing with these as he row, the amount is found in the compartment or square acquires facility and certainty of expression. beneath the one and opposite the other. Thus, 2 times 2 are 4; 5 times 6 are 30; 12 times 12 are 144. The multiplying of numbers beyond 12 times 12 is There are four elementary departments in arithmetic usually effected by a process of calculation in written -Addition, Multiplication, Subtraction, and Division. figures. The rule is to write down the number to be Addition, multiplied, called the multiplicand; then place under Addition is the adding or summing up of several it, on the right-hand side, the number which is to be numbers, for the purpose of finding their united the multiplier, and draw a line under them. For examount. We add numbers together when we say, 1 ample, to find the amount of 9 times 27, we set down and 1 make 2; 2 and 2 make 4; and so on. The me the figures thus thod of writing numbers in Addition, is to place the 27 (Multiplicand.) figures under one another, so that units will stand under 9 (Multiplier.) units, tens under tens, hundreds under hundreds, &c. 243 (Product.) Suppose we wish to add together the following numbers—27, 5, 536, 352, and 275; we range them in Beginning with the right-hand figure, we say 9 times columns one under the other, as in the margin, and 7. are 63; and putting down 3, we carry 6, and say 9 draw a line under the whole. "Beginning at the lowest times 2 are 18, and 6 which was carried makes 24; and figure of the right-hand column, we say 5 and writing down these figures next the 3, the product is 27 found to be 243. 2 are 747 and 6 are 13–13 and 5 are 18-18 and 7 are 25; that is, 2 tens and 5 units. We 5463 When the multiplier consists of two or more 536 now write the 5 below the line of units, and 34 figures, place it so that its right-hand figure 352 carry or add the 2 tens, or 20, to the lowest comes exactly under the right-hand figure of 21852 275 figure of the next column. In carrying this 16389 the multiplicand; for instance, to multiply 1195 20, we let the cipher go, it being implied by 5463 by 34, we proceed as here shown. Here the position or rank of the first figure, and take only product of which being written down, we proceed to 185742 the number is multiplied, first by the 4, the the 2; we therefore proceed thus—2 and 7 are 9-9 and 5 are 14–14 and 3 are 17–17 and 2 are 19. multiply by 3, and the amount produced is placed below Writing down the I, we proceed with the third column, the other, but one place farther to the left. 76843 A line is then drawn, and the two procarrying 1, thus-1 and 2 are 3-3 and 3 are 6-6 and 5 are îl. No more figures remaining to be added, 4563 ducts added together, bringing out the both these figures are now put down, and the amount 230,529 result of 185742. We may in this man 4,610,58 or sum of them all is found to be 1195., Following ner multiply by three, four, five, or any 38,421,5 this plan, any quantity of numbers may be summed number of figures, always placing the pro 307,372 up. Should the amount of any column be in three duct of one figure below the other, but figures, still, only the last or right-hand figure is to be 350,634,609 shifting a place farther to the left in each put down, and the other two carried to the next column. line. An example is here given in the down the 7 and carry the other two figures, which are thus, 3 x 8 = 24, signifies, that by multiplying 8 by For example, if the amount of a column be 127, put multiplying of 76843 by 4563. Multiplication is denoted by a cross of this shape x : 12; if it be 234, put down the 4 and carry 23. For the sake of brevity, in literature, addition is often 3, the product is 24. A number which is produced by denoted by the figure of a cross, of this shape +. the multiplication of two other numbers, as 30 by 5 and Thus, ? + 6 means 7 added to 6; and in order to ex; The 5 and 6, called the factors (that is , workers or 6, leaving nothing over, is called a composite number. press the sum resulting, the sign : which means equal to, is employed, as 7 + 6 13; that is, 7 and 6 are agents), are said to be the component parts of 30, and equal to 13. Again, 8 + 5 + 9 22. 30 is also said to be a multiple of either of these num bers. The equal parts into which a number can be Multiplication. reduced as the twos in 30—are called its aliquot parts. Multiplication is a short method of addition under A number which cannot be produced by the multiplicacertain circumstances. If we wish to ascertain the tion of two other numbers, is called a prime number. amount of twelve times the number 57, instead of When the multiplicand and multiplier are the samesetting down twelve rows of 57, and adding them to that is, when a number is multiplied by itself oncegether, we adopt a shorter plan, by which we come to the product is called the square of that number: 144 the same conclusion. For ascertaining the amount of is the square of 12. all simple numbers as far as 12 times 12, young persons Subtraction. commit to memory the following Multiplication Table, a knowledge of which is of great value, and saves much from a greater, to find what remains, or the difference Subtraction is the deducting of a smaller number trouble in after life : between them. We subtract when we say, take 3 from 5, and 2 remains; 4 from 10, and 6 remains. To ascer3 tain what remains, after taking 325 from 537, we 6 | 8 | 10 537 proceed by writing the one under the other, as 325 here indicated, and then subtracting. Commenc6 | 9 12 15 212 ing at 5, the right-hand figure of the lower and 4 | 8 | 12 16 | 20 24 | 28 smaller number, we say, 5 from 7, and 2 remains; setting down the 2, we say next, 2 from 3, and I re5 10 15 20 mains; and setting down the 1, we say, 3 from 5, and 2 remains; total remainder, 212. 7 14 28 | 35 To subtract a number of a higher value, involving the carrying of figures and supplying of tens, we pro ceed as in the margin. Commencing as before, we find 9 | 18 that 5 cannot be subtracted from 2, and therefore 8432 supply or lend 10 to the 2, making it 12; then we 10 20 | 30 40 50 60 | 70 6815 say, 5 from 12, and 7 remains. Setting down the 22 33 | 44 55 66 77 1 88 1617 7, we take 1, being the decimal figure of the number which was borrowed, and give it to the 1, 24 72 | 84 | 96 120 | 132 | 144 making it 2, and taking 2 from 3, we find that I remains, 1 2 4 5 6 7 8 9 10 11 12 2 2 12 14 16 18 |