the assertion to be made is true of the individuals, the plural should be used; in most instances the plural alone is right, in few or none is it altogether wrong. 2. Singular nouns (or pronouns) coupled by and naturally require the verb and pronoun to be in the plural; as, 'John and William are learning their lessons.' One might think that no doubt could ever arise in such constructions; yet, if we follow the rule blindly, according to the letter, and not the spirit, we might go wrong even in this simple case. The following sentences are correct, and yet, if we looked only at the grammatical form of the subjects, the verbs should be plural: 'The wheel and axle is one of the mechanical powers. Bread and butter is my usual breakfast. Hanging and beheading is in that country the punishment of treason.' The grounds of these exceptions are evident. Instead of and, the preposition with is sometimes used to connect the parts of a collective subject, and then it is a disputed point whether the verb should be singular or plural. Ex., 'The captain with his men were taken prisoners.' The sense requires the plural; but, grammatically, there is only a singular subject, captain, for men is the objective after with; so that by rule it should be, was taken prisoner.' It is better to avoid all such constructions, and say simply: 'The captain and his men,' &c. ; or, if the captain must be kept prominent, 'The captain was made prisoner, along with his men.' Singular nouns connected by or, take the verb in the singular; as, 'Either John or James has the watch in his pocket.' The reason is obvious; it is one or the other, but only one, that has it. A singular verb is also generally used when neither -nor is the connective. Ex., 'Neither John nor James has the watch.' Some, indeed, maintain that it ought to be have, since the 'not-having' is affirmed of both. To this it may be answered that, though affirmed of both, the affirmation is made of each separately-'Neither (has) John nor has James.' This is an effect of all the distributive adjectives, each, every, either, neither. Ex., 'Every physician, and every clergyman, is by education a gentleman.' Thus the number of the verb is, with a little consideration, not so difficult a matter to determine. But what shall we say about the pronoun in cases like the following: Either the boy or the girl has left his (?), her (?) gloves. Any person can do it for himself (?), herself (?). There seems to be only one way of escape from this difficulty -to use the plural pronoun their, themselves, in which the distinction of gender is not marked. Whenever strict grammar and sense conflict, it is the former that must give way. Examples of this solution of the difficulty might be collected in any number from the best writers. Take the following: Let each esteem other better than themselves' (Bible). 'Every person's happiness depends, in part, upon the respect they (for he or she) meet with in the world' (Paley). Everybody began to have their vexation' (Miss Austen). Concord of Pronouns. The agreement of personal pronouns with the nouns to which they refer, has been noticed along with the concord of verbs. Great care is requisite to make the reference of the pronoun clear and unmistakable. The right use of relative pronouns is an important point in language. Who is employed when the reference is to persons, and which when it is to inferior animals or things. That is applied both to persons and things; and it is usually said that that may be used as a substitute for who and which. But this statement requires considerable modification, as may be made to appear thus: Who and which occur having two very different senses or effects. (1.) In the following examples they introduce adjective-sentences, in order to limit or define nouns. Ex., I met the man whom we saw yesterday. The old house which stood at the corner of the street has been burnt down.' (2.) In other cases they introduce sentences that are either principal, or, if subordinate, of the kind called adverbial; and then the relative itself is always resolvable into a conjunction of some kind and a personal pronoun. Ex., 'I met the gardener this morning, who told me that there had been rain during the night.' There are here two coordinate sentences, and who and he. 'Why should we consult Charles, who (= for he) knows nothing of the matter. He struck the poor dog, which (although it) had never done him harm. He by no means wants sense, which (= but that) only serves to aggravate his folly.' = Now, in no case where the relative is thus resolvable, could that be substituted for who or which; it would alter the sense entirely. It is only when the purpose of the relative clause is to define the thing meant, that that is ever applied; and for this purpose its use is in general preferable to that of who or which. The city that is called Rome, was founded by Romulus,' is an easier, more natural mode of expression than, 'The city which is,' &c. Introduced by that, the relative clause coalesces better with the noun, and its adjective effect is better felt, than with the heavier, less compact connectives. When the relative introduces a defining clause, and is in the objective case, it is often omitted. Thus, it is more idiomatic English to say, 'I have found the book you want,' than-'the book that you want?' GOVERNMENT. One word is said to govern another when it seems to cause it to take on a particular form or inflection, as when a pronoun coming after a preposition takes the objective form-with me (not I). The Possessive Case.-A noun takes the possessive form when another noun follows it in the relation of its property; as, 'The king's crown.' The possessive relation is often more conveniently expressed by of; as, 'The crown of the most powerIf the name of the owner be ful king on earth.' a compound name, the last of the component parts only receives the sign of the possessive: thus, 'The Queen of Great Britain's prerogative;' also when there are two separate names, as, 'Robertson and Reid's office.' The possessive case sometimes stands alone, the governing word being understood. Ex., 'He is at his father's (house). Have you seen St Paul's (Church)?' The Objective Case.-Transitive verbs and prepositions take the objective case after them. It is only when a pronoun is the object that there is 591 No one any change on the word governed. Errors are never made on this point when the pronoun immediately follows the verb or preposition. thinks of saying: 'I saw he, or, 'The man spoke to we.' It is only when the object is at some distance from the governing word, so that the objective relation is obscured, that a wrong case is apt to be used. Ex., He that promises too much, do not trust;' for, Him that,' &c. If the sentence were arranged differently, beginning, 'Do not trust him, &c., the error would never occur. especially in the use of who and whom that errors It is of case are apt to be made. Ex., 'Do you know who you speak to?' for—' whom you speak to.' Care requires to be taken not to use the objective case when the pronoun is the subject of a verb. Ex., 'Who did that? Me.' By supplying the ellipsis, we see that the nominative should be used-1 (did it). There is such a strong propensity in all, learned and unlearned, to use me, him, them, &c. in answer to questions and in such phrases as, 'It is me, it was her,' &c. that there are not wanting those that defend the practice as right, and according to the instinctive genius of the English language. They hold that there are two forms of the nominative case, one to be used when a verb immediately follows, and another when the pronoun has to stand alone. analogy of the French language is in favour of this view; for while 'I am here,' is 'Je suis ici,' the answer to 'Who is there?' is 'Moi' (me); and 'C'est moi' (it is me), is the legitimate phrase, never 'C'est je' (it is I). But as yet this opinion, however much countenanced by usage, is considered a heresy by orthodox grammarians; and, accordingly, such expressions as, 'It was her, I am taller than him,' are to be avoided as errors. The It is the dread of falling into this error that drives many into the very opposite of using the nominative case after a verb or preposition, in such expressions as: 'Between you and I' Apposition. When one noun (or pronoun) is joined to another to explain it, the two are said to be in apposition; and then they must be both in the same case. Ex., London, the capital, is the greatest city in the world. They hanged the leader of the gang, him that had so long defied the law. The leader of the gang was hanged, he that had so long defied the law.' The verb To Be.-The verb to be has the same case after it that it has before it. Ex., 'Alfred was king Here Alfred is nominative to was, and as was merely declares that king is another name for the same person, king agrees with Alfred in case. 'It was he. He is here in the nominative, because it is the nominative to was; in, 'I took it to be him,' him is objective, because it is the object to the verb took. tion is suppressed. Ex., Had I known that,' by there, here, where. Ex., 'There stood a church 2. The transitive verb precedes its object. 3. The adjective immediately precedes the noun. Excep.-When the adjective has any words be neglected? joined with it. Ex., 'A question too important to qualifies; with an intransitive verb it is placed after. 4. The adverb is placed before the adjective it Ex., 'A very good man. She dances well. When the verb has an object, the adverb usually follows it; as, 'The ball wounded him severely. But no general rule can be given regarding the adverb, except to take care that it be so placed as to affect the word it ought to affect. More errors in this respect are committed with the adverb only, than with any other. According to the position of only, the very same words may be made to express several very different meanings. Ex., (1.) He only lived for their sakes. (2.) He lived only for their sakes. (3.) He lived for their sakes only. (4) He lived for their sakes alone. These sentences imply respectively— work, did not die, &c. for their sakes. (1.) He lived for their sakes, but did not (2.) He lived for their sakes, and not for any other reason. more worthy reason. sake of any other persons. In observed in plain, unimpassioned prose. application in syntax. mystery of how to use commas, semicolons, &c. Under the head of Syntax it is usual to give a number of rules about the choice of words and phrases. On this vast field we cannot enter. It involves the knowledge of the correct meaning of words, in the language; and belongs rather to all the words, and idiomatic combinations of is acquired only by extensive experience-by readLexicography than to Grammar. This knowledge ing good authors and hearing good speakers. ARITHMETIC-ALGEBRA. In and y not have received mines independent of st numbers, havd N the present and succeeding sheets, an attempt | in a convenient manner. The had the benefit of a regular course of instruction in the subject, 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 recog- | nising 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 from them to us and all civilised nations. are one (or the unit by itself), two (or a unit more than one), three, four, five, six, seven, eight, nine. The number following is called ten. It is regarded as a new unit, or a unit of a new order. Then we count by tens, as we do by units, namely, one ten, two tens or twenty, three tens or thirty, forty, fifty, sixty, seventy, eighty, ninety. Between ten and twenty there are nine other numbers, namely, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen. Between twenty and thirty there are nine numbers, and also between thirty and forty, forty and fifty, &c. In this way we arrive at ninety-nine, and the addition of one to this gives one hundred. We reckon by hundreds as we do by tens, thus: one hundred, two hundreds, three hundreds, &c. Then by placing successively between the words one hundred and two hundred, between two hundred and three hundred, &c. the names of all the numbers from one up to ninety-nine, we have at last the number nine hundred and ninety-nine, and the addition of one to this gives ten hundred or one thousand. This, again, is taken as a new unit, and we say one thousand, two thousand, three thousand, &c. ten thousand nine hundred and ninety-nine thousand. The number nine hundred and ninety-nine thousand nine hundred and ninetynine can thus be formed, and the addition of a unit to this gives ten hundred thousand, or a thousand thousands, or one million. In the same manner a million millions may be formed, and this is called a billion, &c. ... Numeration written.-The invention of arithmetical notation must have been coeval with the earliest use of writing, whether hieroglyphic or otherwise, and must have come into use about the time when it was felt that a mound, pile of stones, or huge misshapen pillar, was insufficient as a record of great events, and required to be supplemented by some means which would suffice to hand down to posterity the requisite information. The most natural method undoubtedly was to signify 'unity' by one stroke, thus: ; 'two' by two strokes, II; 'three' by three strokes, III, &c.: and, as far as we know, this was the method adopted by most of those nations who invented systems of notation for themselves. It is shewn on the earliest Latin and Greek records, and is the basis of the Roman, Chinese, and other systems. We have thus a convenient division of the different notational systems into the natural and artificial groups, the latter including the systems of those nations who adopted distinct and separate symbols for at least each of the nine digits. The Roman and Chinese systems are the most important of the former, and the Hebrew, later Greek, and 'decimal' systems of the latter group. Roman System.-The system adopted by the Romans was most probably borrowed at first from the Greeks, and was distinguished equally by its bers. distasteful to so fastidious a race, and they hit B. Units. ... 2 3 ....5 (introduced)..6.. 0 or ..... ... 7. 20 ..70 80 .. W. Hundreds. represents. .....100 200 300 ...400 ..500 .600 700 .800 95 or 4 (introduced).90, A. (introduced) 900 simplicity and its cumbrousness. The following seems to be the most probable theory of its development. A simple series of strokes was the basis of the system; but the labour of writing and reading large numbers in this way would soon suggest methods of abbreviation. The first and most natural step was the division of the strokes into parcels of tens, thus, II, a plan which produced great facility in the reading of numbers. The next step was to discard these parcels of ten strokes each, retaining only the two cross strokes, thus, X, as the symbol for 10. same method as larger numbers came to be used, Continuing the they invented a second new symbol for 100, thus, & represents....I C(which was at first probably the cancelling stroke for ten Xs, in the same way as X was originally the cancelling stroke for ten units); and for the sake of facility in writing, subsequently employed the letter C, which resembled it, in its place. The circumstance that C was the initial letter of the word centum, 'a hundred,' was doubtless an additional reason for its substitution in place of the original symbol for 100. An extension of the same process produced M, the symbol for 1000, which was also written A, M, and very frequently CIO. This symbol was probably suggested by the circumstance that M was the initial letter of the Latin word mille, signifying a thousand. The early Roman system went no higher. But though the invention of these three symbols had greatly facilitated the labour of writing down and reading off numbers, further improvements were urgently required. The plan of 'bisection of symbols' was now adopted; X was divided into two parts, and either half, V or A, used as the symbol for 5; was similarly divided, F, or L, standing for 50; and N, CI, or I, was obtained in the same manner, and made the representative of 500. The resemblance of these three new symbols to the letters V, L, and D, caused the substitution of the latter as the numerical symbols for 5, 50, and 500. A final improve ment was the substitution of IV for 4 (in place of IIII), IX for 9 (in place of VIIII), XC for 90 (instead of LXXXX), and similarly XL for 40, CD for 400, CM for 900, &c. ; the smaller number, when in front, being always understood as subtractive from the larger one after it. This last improvement is the sole departure from the purely additional mode of expressing numbers; and if the symbols for 4, 9, 90, &c. be considered as single symbols, which they practically are, the deviation may be looked upon as merely one of form. In later times, the Roman notation was extended by a multiplication of the symbol for 1000, thus CCI represented 10,000; CCCɔɔ represented 100,000, &c.; and the bisection of these symbols gave them IƆ and ɔɔɔ as representative of 5000 and 50,000 respectively. This, in all probability, is the mode according to which the Roman system of notation was constructed. To found a system of arithmetic upon this notation would have been well-nigh impossible; and so little inventive were the Romans, that the attempt seems never to have been made. They performed what few calculations they required by the aid of the Abacus. M could be expressed, but by putting a mark, called By these symbols, only numbers under 1000 iota, under any symbol, its value was increased a thousandfold, thus = 1000, 20,000; or by subscribing the letter M, the value of a symbol was raised ten-thousandfold, thus,=80,000. For these two marks, single and double dots placed over the symbols were afterwards substituted. This improvement enabled them to express with facility all numbers as high as 9,990,000, a range amply sufficient for all ordinary purposes. Furby Apollonius, who also, by making 10,000 the root of the system, and thus dividing the symbols ther improvements were made upon this system into tetrads, greatly simplified the expression of very large numbers. Both Apollonius and Archimedes had to a certain extent discovered and depending on their position and multiplicative of their real value, but this principle was applied to employed the principle of giving to symbols values tetrads or periods of four figures only, and the multitude of symbols seems to have stood in the way of further improvement. who was the chief improver of the system, discarded all but the first nine symbols, and applied Had Apollonius, the same principle to the single symbols which he anticipated the decimal notation. applied to the 'tetrad' groups, he would have The Decimal System.-The first nine numbers are represented by nine separate symbols, or figures, known as the Arabic numerals; namely: I, 2, 3, 4, 5, 6, 7, 8, 9. one, two, three, four, five, six, seven, eight, nine. The nine figures, with the addition of the symbol o, enable us to write any number, however large or however small it may be. we suppose these figures to represent not only simple units, but units of any order whatever ; In the first place, 3 may represent three simple units, three tens, three hundreds, &c.; and so with the other figures. the figures then represent units of any order, we must see when they represent units of a particular As order. Taking several columns, as is here shewnGreek System.-The Greeks at first used a placed in the column nearest to the right, the tens and let it be understood that the simple units are method similar to the Romans, though at the in the column next on the left, the hundreds in same time they appear to have employed the the column next again, and so on-then we may letters of the alphabet to denote the first 24 num-write down any number; as, for example, three 594 thousand five hundred and forty-eight, by placing | left, denotes tens; the 3 denotes hundreds; and the Tens of thousands. Thousands. Hundreds. Simple units. 3548 79 each of the units in the column allotted to that particular unit; thus the 3 is placed in the column of thousands, the 5 in that of hundreds, the 4 in that of tens, and the 8 in the unit column. Similarly with the number two thousand and seventy-nine, the 2 is placed in the thousands column, the 7 in that of tens, and the 9 in the unit column. In this way we can write down any number; but we may, by a simple device, save ourselves the trouble of drawing the lines to form the columns ; for, let it be understood, that the simple units are put on the right, the tens next on the left, the hundreds next on the left of the tens, the thousands on the left of the hundreds, the tens of thousands on the left of the thousands, and so on-then we may write the number three thousand five hundred and forty-eight at once, thus: 2 Again, the number two thousand and seventy-nine would be written 3548 2 79 As there are here no units of hundreds, their place would be unoccupied, and we would be very apt to take the 2 to represent hundreds, instead of thousands, as it is intended to do. Now, to prevent any mistake arising on this account, we introduce the symbol o or zero to keep the place of the hundreds. This symbol has no value of itself, but merely serves for keeping the places of those units which may be wanting in any number; thus the number two thousand and seventy-nine will be written 2079. Again, two hundred and twelve thousand and eight will be written 212008, because the units of hundreds and also of tens are wanting, and their places are supplied by zeros. The number fifty will be written 50, because the unit figure is wanting. In the same way, five hundred will be written 500. In this manner we are enabled to write down any number. It will at once be evident that if we annex a zero to the right of any number, we multiply it by 10, for by doing so we remove each of the figures one place to the left. Similarly, by annexing two zeros, we multiply by 100; annexing three zeros, we multiply by 1000; and so on. Decimal Numbers.-In considering the number 5555, the 5 nearest to the right denotes units; the next on the left, tens; the next again, hundreds; and the last on the left, thousands. Or proceeding from left to right, we see that the figure on the right is ten times less than the one immediately on its left; thus the 5 on the left denotes thousands; the next on the right, hundreds; the next again, tens; and that on the left, units. There is no reason, however, why this last figure should not in its turn be ten times greater than the figure immediately on its right; and this, again, ten times greater than the next on the right; and so on. By thus extending the decimal system, we will be enabled to represent all fractions, however small. For example, if we take the following expression, 43276918, in which the unit figure, to mark its position, is underlined, we can at once tell the value of each figure; thus 7 being the unit; 2, the first on the 4, thousands. Again, 6, the first figure on the right of the unit, being of ten times less value than if it stood in the units' place, denotes tenths; the 9, the next on the right, denotes hundredths; the 1, thousandths; and the 8, tenths of thousandths. Knowing the position of the unit figure, then we at once know the value of any other figure. The unit figure is marked, not by underlining it, but by placing a point or dot immediately on its right. The above number would be written 4327.6918, The dot is called the decimal point, and separates the whole number in the expression from the fractional part of it. We are also enabled to read off at once any decimal expression; thus 5 is read as decimal 5 or 5 tenths, ·05 as 5 hundredths, .005 as 5 thousandths. Again, 1.5 is read 15 tenths, 1.15 as 115 hundredths-that is, we read off the number as if it contained no decimal point, and give to it the denomination of the last figure on the right. It is evident that a decimal number is multiplied by 10, by 100, by 1000, &c. by simply moving the point one, two, three, &c. places to the right. Thus the number 3.4568, when multiplied by 10, becomes 34.568; when multiplied by 100, becomes 345-68; and so on. In the same manner a number is divided by 10, by 100, by 1000, &c. by simply moving the point one, two, three, &c. places to the left; thus the number 3.4568, when divided by 10, by 100, by 1000, becomes 34568, 034568, 0034568. When we simply suppress the point in a decimal expression, we multiply it by 10, 100, &c. Thus, if for 3.456 we write down 3456, we have multiplied the number by 1000, for the point is now supposed to be on the right of the 6. ADDITION AND SUBTRACTION OF WHOLE OR DECIMAL NUMBERS. The two fundamental operations in arithmetic are addition and subtraction, as a number can only be altered by increase or diminution. Addition is that process by which we increase a number by a given number: or it is that process by which we form into one number several other numbers. The resulting number is called the sum. It is easy to add to a given number another small number; thus, to add 4 to 5, it is sufficient to add one unit to the 5 as often as there are units in 4; thus we say 5 and I make 6, 6 and 1 make 7, 7 and I make 8, 8 and I make 9; and 9 is the sum required. In the same way we might add to a number another number consisting of two or more figures; but it is evident that this process would be very tedious, and when the number to be added was very large, it would be almost impracticable; and hence we consider each of the numbers to be added as being resolved into its component units, and written one under the other, in such a way that the units of the same order stand under each other; we then add the several units separately. Thus let it be required to add together 453, 7546, 5314, 208. Here the number 453 is resolved into 3 simple units, 5 tens, and 4 hundreds. Again, 7546 is resolved into 6 simple units, 4 tens, 5 hundreds, and 7 thousands; and so with the 595 |