sense. Good writers endeavour to avoid requiring | words fall from her mouth. Coleridge somewhere either parenthetic marks or dashes, both of which indicate irregularities of thought and expression. CONCLUSION. tells that he was once much prepossessed in favour of an individual whom he met at a dinner-table, and who appeared a dignified and respectable person, until, some kind of fruit being introduced, he heard him exclaim, Oh, them's the jockies for me!' Words are the exponents of conditions of mind, and when mean ones are used, we unavoidably suppose the condition of mind to be mean. ERRORS IN PRONUNCIATION. We have now explained the Etymology and Syntax of the English tongue, as far as our limits permit; and in drawing to a close, we may be allowed to impress on our readers the value of the science which we have been endeavouring to expound. If they have intelligently gone along with us in our various remarks, 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 to recognise the different modifications that words undergo, and the different arrangements of which they are susceptible, to express difference of thought, we have exercised many of the mental faculties, and in so far laid the foundation of what is much wanted-a just system of Logic. The sources whence the student will derive effectual aid in the prosecution of this interesting subject, we have already pointed out incidentally; but let no one lament too much though he should not have access to them. Rather let him, by additional thought on his own part, make up for the deficiency, and he may rest assured that, by accustoming himself to mark the different modes of expression he meets with in reputable authors, a system of grammar will evolve itself, which will be all the more valued-if we may not say valuable -that it has been wrought out by his own exertions, and not received by tradition or passively from the hands of another. Following this plan, the real method of induction, he will either reproduce the rules which we have set before him, or else see their erroneousness. So that, in either case, we shall deserve well of him; for, if we are right in anything, we shall have served as a guide to him; and in those points where we have erred, we shall have put him on the way to find out our errors. We know very well that the pupil cannot see with our eyes, and we have therefore only endeavoured to direct his attention to such objects as he may see with his own. So far as he sees, he should believe, and no farther. To dogmatise is the method of a grammatist, but our ambition has been to act the part of a philosophical grammarian, and, as such, we cannot conclude without warning our readers never to forget that words in themselves are nothing, and that they are only valuable in so far as they are the symbols of ideas. Beautifully and justly has Johnson said, Words are the daughters of earth, and things only are the sons of heaven.' Language is but a vehicle of thought, or, at best, its instrument, and to view it as an ⚫end unto itself,' is the vain humour of a pedant. Let none be so taken up with words as to forget solid things. mmmm COMMON ERRORS CORRECTED. The sound instead of g at the ends of words. Examples-Somethink, nothink. The addition of r at the ends of words ending in vowels. Examples-Idear, windor, Elizar. Changing the termination en, ain, or eign, into ing; as garding for garden, founting for fountain, sovering for sovereign, and the like. UNGRAMMATICAL FORMS. Between you and I, there is a great want of conscientiousness in most partisans. Correction-Between you and me, &c. I am not so proud as him. Cor.-As he. While the house was being built. Cor.-While the house was in the course of being built. He don't go to town to-day. Cor.-He does not go to town to-day. I rather think he is out of town. Cor.-I believe he is out of town. I had better go myself. Cor.-It were better that I should go myself. I had oblige to go. Cor.-I was obliged to go. John is tall in comparison to James. Cor.-John is tall in comparison with James. He is a very rising man. Cor. He is rising very rapidly. She readied a dish for us. pared, a dish for us. Cor. She cooked, or pre The then Earl of Winchelsea; the then Mrs Bennet. Cor.-The Earl of Winchelsea of that time; the Mrs Bennet then living. He lays asleep in the cabin. Cor.-He lies asleep in the cabin. His health was drank. Cor.-His health was drunk. The dinner was all eat up. Cor.-The dinner was all eaten up. I went to table and eat very heartily. Cor.-I went to table and ate very heartily. A couple of shillings. Cor.-Couple can only be properly applied to objects in connection; as, a married couple, a couple of pointers. The remaining space of the present sheet could scarcely, we think, be better employed than in enumerating some examples of the most common errors in the pronunciation and selection of words. In every part of the country there are some peculiar vices of speech, which have been handed down from one generation to another, and are generally so inveterate in most minds, from the effect of early habit, that no cultivation which the mind may receive in mature life altogether obliterates them. For any one who has occasion to mix in refined society to be thus liable 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 John, James, and Robert, were sober workmen, the latter particularly so. Cor. The last particularly so (the objects enumerated being more than two). Ask at him. Cor.-Ask him. I could not give him credit, without he changes his behaviour. Cor.-I could not give him credit, unless | He 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 have 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. I took some cream into a bowl. Cor.-I took some cream in a bowl. I am going for to do it. Cor.-I am going to do it. He was a devoted antiquarian all his days. Cor.He was a devoted antiquary all his days. (Antiquarian is the adjective.) James is going to be a medical man. is going to be a physician, surgeon, or titioner. Cor.-James medical prac I don't know, but I will inquire at my friend. Cor. -Of my friend. I was calling for you yesterday. Cor.-On you yesterday. Oh, I will fall, and nobody shall help me. Cor.-Oh, I shall fall, and nobody will help me. I have been to London, and am now going for Liverpool. Cor. I have been in London, and am now going to Liverpool. He was married on Miss Edmonstone. Cor. He was married to Miss Edmonstone. They were some distance from home when the accident happened. Cor. At some distance, &c. He lives opposite the Royal Exchange. Cor.-Opposite to, &c. Pray, sit into the fire. Cor.-Pray, sit near the fire. The performance was approved of by all who understood it. Cor. The performance was approved by all. They attacked Northumberland's house, whom they put to death. Cor. They attacked the house of Northumberland (or the Duke of Northumberland), whom they put to death. It is true what he says, but it is not applicable to the point. Cor.-What he says is true, &c. Together with the national debt, the greatest national advantages are also transmitted to succeeding generations. Cor.-Also is superfluous. Failing in his effort, he again repeated it. Cor.Again is superfluous. He is noway thy inferior, and in this instance is noways to blame. Cor. He is in nowise thy inferior, and in this instance is not at all to blame. It is neither more nor less than medicine in disguise. Cor. It is simply medicine in disguise. The master never challenged him for stealing. Cor. -The master never reproved him for stealing. He charged me with want of resolution, in which he was greatly mistaken. Cor.-He charged me with want of resolution, but in this censure he was greatly mis taken. He gave her a beautiful book in a present. Cor. There was a quantity of people present. Cor.-There was a number of people present. It is above a year since the time that I left school. Cor. It is more than a year since I left school. He felt the peculiarness of his situation. Cor.-He felt the peculiarity of his situation. In like manner delicacy should be preferred to delicateness, incapability to incapableness, &c. He was guilty of such atrocious conduct, that he was deserted by his friends for good and all. Cor.-He was guilty of conduct so atrocious, that he was entirely deserted by his friends. it. OBSOLETE, AWKWARD, AND MEAN FORMS. I had as lief do it myself as persuade another to do Cor.-I would as readily, &c. He convinced his opponent by sheer dint of argument. Cor.-Entirely by force of argument. He is not a whit better than those whom he so liberally condemns. Cor. He is not in any degree, &c. He stands upon the bond, and will not abate a jot of his claim. Cor.-He insists on the strict terms of the bond, and will not in the least abate his claim. Good satin, I take it, is considerably superior to common silk. Cor.-I presume, &c. You have no call to do it. Cor.-You have no occasion to do it. I have no right to pay. Cor.-I am not bound to pay. Politics too often sets men by the ears. When they come to words, and fall out, reason is generally lost sight of. I should not wonder but on this occasion there might be broken heads going. Cor.-Politics too often cause quarrels. When men enter into controversy, and differ violently, reason is generally lost sight of. I should not wonder but on this occasion they might commit some violence on each other. We shall have a regular break-up in the ministry. Cor.-We shall have a dissolution of the ministry. He was very dexterous in smelling out the designs of his neighbours. Cor.-In penetrating, &c. He is a thorough-paced knave. Cor.-He is a great knave. Heretofore Hannibal had carried all before him; wherefore he had become very proud, listening to no advice whatsoever; whereas Scipio invariably took counsel from the most sagacious of his officers.-The words in Italics are all obsolete and objectionable. do. He wist not what to do. Cor.-He knew not what to He little wots of the storm that is brewing. Cor.He is not aware, &c. Topsy-turvy, pell-mell, hurly-burly, having a month's mind for a thing, currying favour with a person, dancing attendance on customers, get into a scrape, come to the scratch, flare up, fork out, walk into him, kick up a row, raise a rumpus, and the like-All objectionable from their meanness. We are at one on the slave question. I happen to have a little leisure upon my hands. He might have perceived it with half an eye. My father left this morning by the mail. Cor.- My father went away this morning, &c. When are you to leave?' is in like manner vicious. The place or thing left should always be stated. Slang phrases of all kinds should be received warily. The least objectionable are those which merely suggest comical ideas; those which tend to present light and jocular views of moral error are particularly detestable. It will be the aim of a well-bred and judicious person to make his discourse neither too nice and formal, nor too loose and homely, but, as far as possible, to preserve a medium between the select language employed in literature, and the familiar, and perhaps temporary, phraseology which prevails in ordinary society. ARITHMETIC-ALGEBRA. In the present and succeeding sheet, an attempt is made to convey to the comparatively unlearned mind some knowledge of Mathematical science, both as regards measurement by numbers (ARITHMETIC) and measurement of dimensions (GEOMETRY). The sketch we offer of each is necessarily brief and imperfect; but our end will be gained if we afford that amount of information on the subject which is generally possessed by persons of moderately well-cultivated intellect. A recognition of the value of numbers is coeval with the dawn of mental cultivation in every community; but considerable progress must be made before methods of reckoning are reduced to a regular system, and a notation adopted to express large or complex quantities. An inability to reckon beyond a few numbers is always a proof of mental obscurity; and in this state various savage nations have been discovered by travellers. Some are found to be able to count as far as five, the digits of the hand most likely familiarising them with that number; but any further quantity is either said to consist of so many fives, or is expressed by the more convenient phrase, a great many. Among the North American Indians, any great number which the mind is incapable of distinctly recognising and naming is figuratively described by comparing it to the leaves of the forest; and in the same manner the untutored Negro of Africa would define any quantity of vast amount by pointing to a handful of sand of the desert. On the first advance of any early people towards civilisation, it would be found impossible to give a separate name to each separate number which they had occasion to describe. It would therefore be necessary to consider large numbers as only multiplications of certain smaller ones, and to name them accordingly. This is no doubt what gave rise to classes of numbers, which are different in different countries. For instance, the Chinese count by twos; the ancient Mexicans reckoned by fours. Some counted by fives, a number which the fingers would always be ready to suggest. The Hebrews, from an early period, reckoned by tens, which would also be an obvious mode, from the number of the fingers of the two hands, as well as of the toes of the two feet. The Greeks adopted this method; from the Greeks it came to the Romans, and by them was spread over a large part of the world. NOTATION. Notation is the method of expressing numbers by means of certain signs or figures. The representation of numbers by written signs is an art generally believed to have taken its rise after the formation of alphabets. One of the earliest sets of written signs of numbers of which we have any notice, is certainly the series of letters of the Hebrew alphabet which was used by that people-Aleph, beth, gimel, daleth, he, vau, zain, cheth, teth, standing respectively for the numbers one, two, three, four, five, six, seven, eight, nine. The Greeks directly adopted this plan from the Hebrews, forming their numbers thus:-1 alpha, 2 beta, 3 gamma, 4 delta, 5 epsilon-here, having no letter corresponding with the Hebrew vau, they put in the words anμov Bau to denote six; after which they proceeded with 7 zeta, 8 eta, &c. Before adopting this plan, they had indicated one by iota, probably because it was the smallest of their letters; five by II (P), being the first letter of pente, five; ten by A (D), being the initial of deka, ten. After having for some time adopted the Hebrew mode, they divided their alphabet into three classes: the first ten letters expressing the numbers from one to ten; while twenty, thirty, forty, and so on up to a hundred, were signified by the next nine, No. 88. ninety being expressed by a figure formed on purpose, and resembling the Arabic 5 inverted. The remaining seven letters expressed respectively 200, 300, 400, 500, 600, 700, 800; and for 900 there was another inverted figure. Larger numbers were represented by letters accented in various ways. The Romans, from an early period, had a method of expressing numbers, which seems to have been at first independent of the alphabet. The following intelligible account of it has been given by Professor Playfair: To denote one, a simple upright stroke was assumed; and the repetition of this expressed two, three, &c. Two cross strokes X marked the next step in the scale of numeration, or ten; and that symbol was repeated to signify twenty, thirty, &c. Three strokes, or an open square, were employed to denote the hundred, or the third stage of numeration; and four interwoven strokes M, sometimes incurved M, or even divided CIO, expressed a thousand. Such are all the characters absolutely required in a very limited system of numeration. The necessary repetition of them, however, as often occasionally as nine times, was soon found to be tedious and perplexing. Reduced or curtailed marks were therefore employed to express the intermediate multiples of five; and this improvement must have taken place at a very early period. Thus five itself was denoted by the upper half V, and sometimes the under half, of the character X for ten; L, or the half of the mark for a hundred, came to represent fifty; and the incurved symbol M, or CIO, for a thousand, was split into 10, to express five hundred. These important contractions having been adopted, another convenient abbreviation was introduced. To avoid the frequent repetition of a mark, it was prefixed to the principal character, and denoted the effect by counting backwards. Thus instead of four strokes, it seemed preferable to write IV; for eight and nine the symbols were IX and IX; and ninety was expressed by XC. This mode of reckoning by the defect was peculiar to the Romans, and has evidently affected the composition of their numerical terms. Instead of octodecem [eight and ten-for eighteen], and novemdecem [nine and ten-for nineteen], it was held more elegant, in the Latin language, to use undeviginti [one from twenty], and duodeviginti [two from twenty]. But the alphabetic characters now lent their aid to numeration. The uniform broad strokes were dismissed, and those letters which most resembled the several combinations were adopted in their place. The marks for one, five, ten, and fifty, were respectively supplied by the letters I, V, X, and L. The symbol for a hundred was aptly denoted by C, which had originally a square shape, and happened, besides, to be the initial of the very word centum. The letter D was very generally assumed as a near approximation to the symbol for five hundred; and M not only represented the angular character for a thousand, but was likewise, though perhaps accidentally, the first letter of the word mille.'-Edin. Rev. No. xviii. p. 193. The Hebrew, improved Grecian, and Roman numerals were perhaps sufficient to express any single number with tolerable precision; but it is easy to see that they must have been nearly unfitted for use in the processes of arithmetic. The Greeks certainly contrived to overcome many obstacles in the business of calculation, and even could express fractions-though, from a practice of adding from left to right, and ignorance of the plan of carrying tens to the higher places, their problems were at all times awkward and complicated. The Romans, however, careless of old inconveniences, were still more awkwardly situated than 593 It would be impossible to calculate, even by their own transcendent powers, the service which the Arabic numerals have rendered to mankind. NUMERATION. Numeration is the art of numbering—that is, of expressing any number in words. The Arabic numerical signs now generally in use take the following wellknown forms:-1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The first nine of these, called digits or digital numbers, represent, each, one of the numbers between one and nine, and when thus employed to represent single numbers, they are considered as units. The last (0), called a nought, nothing, or cipher, is, in reality, taken by itself, expressive of an absence of number, or nothing; but, in connection with other numbers, it becomes expressive of number in a very remarkable manner. the Greeks. Let any reader just suppose, for instance, even so simple a question as the amount of XLVIII added to XXXIV! It is evident that placing the 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 mental calculation would not serve, to resort to a mechanical process for performing problems in arithmetic. A box of pebbles called loculus, and a board called abacus, constituted their means of calculation; and of these every schoolboy, and many other persons, possessed a set. The word calculation claims no higher descent than from calculus, a stone or pebble. The board was divided from the right to the left hand by upright columns, on which the pebbles were placed, to denote units, tens, hundreds, thousands, &c. The labour of counting and arranging the pebbles was afterwards sensibly abridged by drawing across the board a horiThe 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 7,8 9 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 interlarge 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 followthemselves 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 from the Arabic word tsaphara, denoting empty or void, this term came afterwards to express, in general, any secret mark. Hence in more troublous times than the present, a mode of writing was practised, by means of marks previously concerted, and called writing in cipher. The Arabic characters occur in some arithmetical tracts composed in England during the thirteenth and fourteenth centuries, particularly in a work by John of Halifax, or De Sacrobosco; but another century elapsed before they were generally adopted. They do not appear to have settled into their present forms till about the time of the invention of printing. 0 Tens. Units. Thousands. Tens of thousands. Hundreds of thousands. +Millions. Tens of millions. Thousands of millions. Units. Tens. ENGLISH. Tens of thousands. Tens of millions. Ten thousands of millions. Billions. Tens of billions. Hundreds of billions. Units. Hundreds. Thousands. FRENCH. Tens of thousands. Tens of millions. Tens of billions. Hundreds of billions. Tens of trillions. Hundreds of trillions, &c. For practice in Notation and Numeration, the reader should write down large numbers alternately in words and figures; at first assisting himself by the use of commas, but gradually dispensing with these as he acquires facility and certainty of expression. SIMPLE OR ABSTRACT NUMBERS. There are four elementary departments in arithmetic -Addition, Multiplication, Subtraction, and Division. Addition. 27 536 352 275 1195 Addition is the adding or summing up of several numbers, for the purpose of finding their united amount. We add numbers together when we say, 1 and 1 make 2; 2 and 2 make 4; and so on. The method of writing numbers in Addition, is to place the figures under one another, so that units will stand under units, tens under tens, hundreds under hundreds, &c. Suppose we wish to add together the following numbers-27, 5, 536, 352, and 275; we range them in columns one under the other, as in the margin, and draw a line under the whole. Beginning at the lowest figure of the right-hand column, we say 5 and 2 are 7-7 and 6 are 13-13 and 5 are 18-18 and 7 are 25; that is, 2 tens and 5 units. We now write the 5 below the line of units, and carry or add the 2 tens, or 20, to the lowest figure of the next column. In carrying this 20, we let the cipher go, it being implied by the position or rank of the first figure, and take only 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. Writing down the 9, we proceed with the third column, carrying 1, thus-1 and 2 are 3-3 and 3 are 6-6 and 5 are 11. No more figures remaining to be added, both these figures are now put down, and the amount or sum of them all is found to be 1195. Following this plan, any quantity of numbers may be summed up. Should the amount of any column be in three figures, still, only the last or right-hand figure is to be put down, and the other two carried to the next column. down the 7 and carry the other two figures, which are For example, if the amount of a column be 127, put 12; if it be 234, put down the 4 and carry 23. For the sake of brevity, in literature, addition is often denoted by the figure of a cross, of this shape +. Thus, 7+6 means 7 added to 6; and in order to express the sum resulting, the sign: which means equal to, is employed, as 7 + 6 13; that is, 7 and 6 are equal to 13. Again, 8 + 5 + 9 = 22. = This table is so well known, that it is almost superfluous to explain that, when any number in the top row is multiplied by any number in the left-hand side row, the amount is found in the compartment or square 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 usually effected by a process of calculation in written figures. The rule is to write down the number to be multiplied, called the multiplicand; then place under it, on the right-hand side, the number which is to be the multiplier, and draw a line under them. For example, to find the amount of 9 times 27, we set down the figures thus-— 27 (Multiplicand.) 9 (Multiplier.) 243 (Product.) Beginning with the right-hand figure, we say 9 times 7 are 63; and putting down 3, we carry 6, and say 9 times 2 are 18, and 6 which was carried makes 24; and writing down these figures next the 3, the product is found to be 243. 5463 34 21852 16389 When the multiplier consists of two or more figures, place it so that its right-hand figure comes exactly under the right-hand figure of the multiplicand; for instance, to multiply 5463 by 34, we proceed as here shown. Here 185742 the number is multiplied, first by the 4, the product of which being written down, we proceed to multiply by 3, and the amount produced is placed below the other, but one place farther to the left. 76843 A line is then drawn, and the two pro4563 ducts added together, bringing out the 230,529 result of 185742. We may in this man4,610,58 ner multiply by three, four, five, or any 38,421,5 number of figures, always placing the pro307,372 duct of one figure below the other, but 350,634,609 shifting a place farther to the left in each line. An example is here given in the thus, 3 x 8 = 24, signifies, that by multiplying 8 by multiplying of 76843 by 4563. Multiplication is denoted by a cross of this shape x : 3, the product is 24. A number which is produced by the multiplication of two other numbers, as 30 by 5 and The 5 and 6, called the factors (that is, workers or 6, leaving nothing over, is called a composite number. 30 is also said to be a multiple of either of these numagents), are said to be the component parts of 30, and bers. The equal parts into which a number can be reduced-as the twos in 30-are called its aliquot parts. A number which cannot be produced by the multiplication of two other numbers, is called a prime number. When the multiplicand and multiplier are the samethat is, when a number is multiplied by itself oncethe product is called the square of that number: 144 is the square of 12. Subtraction. from a greater, to find what remains, or the difference Subtraction is the deducting of a smaller number between them. We subtract when we say, take 3 from 5, and 2 remains; 4 from 10, and 6 remains. To ascertain what remains, after taking 325 from 537, we 537 proceed by writing the one under the other, as 325 here indicated, and then subtracting. Commenc212 ing at 5, the right-hand figure of the lower and smaller number, we say, 5 from 7, and 2 remains; setting down the 2, we say next, 2 from 3, and 1 remains; and setting down the 1, we say, 3 from 5, and 2 remains; total remainder, 212. To subtract a number of a higher value, involving the carrying of figures and supplying of tens, we proceed as in the margin. Commencing as before, we find that 5 cannot be subtracted from 2, and therefore 8432 supply or lend 10 to the 2, making it 12; then we 6815 say, from 12, and 7 remains. Setting down the 1617 7, we take 1, being the decimal figure of the number which was borrowed, and give it to the 1, making it 2, and taking 2 from 3, we find that 1 remains. |