not power completely to amend.* The meaning of these remarks about fractions will be more fully appreciated by the learner when he comes to study the explanations of fractions which will follow at various places in this treatise.

It is proper to notice that things which can be counted or characterized by number, through the statement of how many there are of them, are not necessarily tangible objects of bodily substance, such, for instance, as pebbles, sheep, houses, &c. The word thing may be taken in a very general sense to include any article, event, extent of time, extent of space, or other object of our conception which can be considered by itself, and also as associated with others like itself. Then a single thing is sometimes spoken of as ONE of its kind of things, and sometimes it is called a UNIT. A group or assemblage of any of those single things, as, for instance, a dozen of them or a million of them, is itself also an object of our conception; and may, when we please, be treated as a single thing, or as a unit, of which we can count two, or three, or five, or any other number. The treating of a number of objects as a "unit," meaning one group, and then expressing by a number how many there are of those ones or single groups, is a process of thought which has very important applications in arithmetic, especially in fractional arithmetic.

Now for giving numerical expression to quantities of things, as, for instance, to quantities of money, of weight, of time, or of length, a particular quantity of the thing to be dealt with is in some way specified, and is called a UNIT of that thing; and to that unit some name, or denomination, is usually given. Thus a pound sterling is a unit of money, an ounce is a unit of weight, an hour is a unit of time, and a foot is a unit of length. The selection of the unit in

As one step towards abating the difficulty and perplexity of language here referred to, and also for the sake of brevity, we may, when we find it convenient, call a fractional numerical expression by the shorter name a fractional numeric; and we may use the name numeric to signify numerical expressions, whether numbers properly so called, such as one, two, three, four,....eleven,....seventeen, dc., or fractional numerical expressions, such as, 43, 7.23, &c. This nomenclature will at once supply us with (what is at present an important desideratum) a word free from ambiguity for what is implied by "number" when used in its extended sense, or a single word which will briefly and precisely express what is often designated habitually by the inconvenient name in four words, "whole number or fraction." Further, it would, if it were once sufficiently established in use, allow of our practically keeping the word "number" for its own proper signification, saving us from being obliged to use or interpret it in some cases as restricted to its proper signification, and in other cases as extended so as to include fractional numerical expressions. It is difficult, however, to make any considerable change in language which is already established in customary use, and in the present treatise no complete reform will be attempted. The effort will be to teach almost exclusively the ordinary nomenclature in common use, and which ought to be made known to all learners of arithmetic in present times; but still every effort will be made to use to the best advantage the best parts of the ordinary modes of nomenclature, and, as far as possible, to avoid or to amend the more faulty parts. In this way we must still be contented sometimes to bear with more or less of the old incongruities and ambiguities in verbal statements; and we must be ready to make adjustments or amendments by mental reservations or by temporary explanations. If a pupil were taught arithmetic from a book in which all objectionable or faulty nomenclature was completely obviated, he might sometimes be at a disadvantage in answering in competitive examinations to examiners using ordinary language, defective perhaps in precision or in perspicuity, and might often in ordinary intercourse with other people be insufficiently prepared for ready communication of ideas,

which a quantity of anything is to be expressed is arbitrary; and usually there are several units established in which quantities of a thing may be numerically expressed. Thus time may be expressed in years, in minutes, or in seconds, taken as units, as well as in hours; weight may be expressed in tons, in hundredweights, or in pounds, as units, as well as in ounces; and so on. Sometimes one unit may be much more convenient than another for some particular purpose. Thus it may be more convenient to express a weight of coals as so many tons than as so many ounces; and it may be convenient to express a quantity of tea in pounds as units, and very inconvenient to express it in tons, or as a fraction of a ton, in such a way as will soon be explained.

The unit having been selected, the quantity required to be expressed is specified by stating the number of the selected units which would make up the quantity, if any number of them would exactly do so; or, if not, then by stating the number of them, and the fraction or portion of an additional unit which would make up the quantity. One of the usual modes by which the magnitude of the fraction or additional part of a unit is specified, either quite exactly or with sufficient nearness to the truth, is by conceiving the selected unit to be divided into any convenient number of equal parts, and taking one of these parts as a subordinate unit by which to measure the fractional part; the quantity of that fractional part being stated exactly, or approximately, by telling the number of the subordinate units that it contains. If, for instance, in order to measure the length of an object, we have selected the inch as the primary unit, and an eighth of an inch as the subordinate unit, and if we find the length of the object to be 7 inches and 3 eighths of an inch, we have got the length specified numerically through means of a known unit of length, the inch. We may, however, regard the result in several different ways. One way is to adhere to the mode of thought by which we have arrived at it, and so to consider it to be a numerical expression consisting of two distinct numbers, 7 of one unit and 3 of a smaller unit of the same kind of thing-namely, length. Another way is to regard it as being 7 inches, and an eighth of a length of three inches; so that in this way no mention nor thought of the subordinate unit, before dealt with, is introduced at all. In both cases the fractional part of an inch, required with the 7 inches to make up the whole length, is specified by means of three components-namely, the numbers 3 and 8, and the name of the selected unit of length, the inch--and so it is denoted as 3 inch. The numerical expression in this is called a fraction, and is sometimes more particularly described as an arithmetical fraction. The whole length then comes to be expressed as 7 inches and 3 inch, or more briefly as 73 inches. Then as, in speaking of 5 inches, we say the number of inches is 5; or as, in speaking of 10 inches, we say the number of inches is 10; so, in speaking of 7 inches, we very naturally slip into saying that the number of inches is 73. The occasional extension of application of the word "number" to fractional expressions in this way, while its primary and specially proper signification is also retained, and while statements about numbers are often made which do not hold good for fractional

numerical expressions,* has been noticed already as tending to ambiguity and perplexity. On the other hand, there are important advantages of convenience and of brevity in our agreeing to speak of the number of inches in a given length as being 65, or of the number of gallons required to fill a cistern as being 5, and in speaking often of fractional numerical expressions as being numbers. In fact, with the existing language available to us, we cannot avoid often using the word number in this extended way. Sometimes, for distinction, the true or proper numbers, such as 3, 4, 5, 17, 22, &c., are called exact numbers, whole numbers, or integers (the word integer being just the Latin word for whole), and they may well be called proper numbers; while such expressions as,, 73, &c., are called fractions, or fractional numbers, or fractional numerical expressions; but for brevity, or through inadvertence, such distinctive names are commonly neglected. Now, there is clearly a departure of our language from adaptation to any precise idea when we talk of a fractional number of times a complete unit, or of a complete unit taken a fractional number of times; and instead of trying to bring out a clear notion from those words themselves, we are just to bear in mind that they are to be interpreted as meaning either a fractional part of a complete unit, as in the case ofor, or else a number of the units and a fractional part of another of them, as in the case of 7.

Often in practice a quantity of any kind of thing is expressed

*In support of this assertion two instances may be cited, one exemplifying usage in practical business affairs, and the other exemplifying usage among scientific writers on arithmetic. These two may suffice, but the list might easily be extended indefinitely.

In the public regulations for Post Office Savings Banks, issued by authority of the Postmaster-General, the intimation is made that "Deposits of one shilling, or of any number of shillings, or of pounds and shillings, may be made by any person at the Post Office Savings Banks," subject to some provisions which need not be considered here, being merely for assigning limits to the amounts which will be accepted as deposits from one person. Now, the words here quoted would convey a false statement of what the Post Office authorities really mean to announce, if the word "number" were allowed to mean a fractional numerical expression. The announcement is obviously framed on the presumption that the word "number" in it can only mean legally what in the present treatise in the text above is stated as its only proper signification-namely, what is commonly designated as "a whole number; as, for instance, 3, 4, 5,....16,....22, and the like; but not 23, 33, 5.17, or the like. If an intending depositor, understanding the word "number" in the extended sense in which it is very often and also quite authoritatively used, would offer a deposit of 3 pounds, his offer would be refused, as it would amount to 31. 12s. 6d., which is not contemplated in the regulations as an amount to be accepted as a deposit.

Again, in the treatise on arithmetic by Professor De Morgan, an author justly recognised as of high authority in arithmetical science, we find the statement (in the chapter on Division, § 99, at page 48 of the 5th edition, 1846) that “All numbers are measured by 1; that is, contain an exact number of units." This statement, again, only holds good if the application of the word "number" to fractional numerical expressions is excluded. Throughout his treatise on arithmetic, indeed, he seems generally to have been very careful to avoid applying the name "number" to fractional numerical expressions. These he generally calls fractions, whether they be greater or less than one. He usually speaks of "numbers" and of "whole numbers" as synonymous. But still occasionally he seems to slip, as others habitually do, nto speaking of a fractional numerical expression under the name "a number," as, for instance, where (at page 82, § 151, of his 5th edition) he speaks of a number of miles as having been measured and found to be 17.846217 miles.

by a number of units of it, each of one magnitude; together with another number of units, each of a smaller magnitude; and perhaps with other numbers of units of still smaller magnitudes added. Thus a quantity of money may be expressed by a certain number of pounds, together with a certain number of shillings, and a certain number of pence, as when we state a quantity of money by the designation £26-15-11; and a period of time may be expressed as 5 hours, 23 minutes, and 18 seconds. Now, the expression £26-15-11 may properly be called a numerical expression for a certain quantity of money; and that numerical expression may properly be said to be made up of three numbers of three distinct units. Any such expression is called a compound expression, and the quantity expressed is often called a compound quantity. The quantity so expressed is also spoken of as being expressed in units of different magnitudes, or expressed in units of different denominations, or, briefly, as being a quantity in different denominations; and the expression is said to comprise different units. When a quantity is expressed by a number of units of it, each of one magnitude, it is sometimes spoken of as a simple quantity, or a quantity in one denomination, or a quantity expressed in units of a single denomination, or, briefly, a quantity expressed in one unit.

A number is said to be abstractly considered when it is considered as disassociated from any particular objects or units which it might count or characterize numerically, and as left applicable to things in general. For instance, when we say twice three are six, or four and three are seven, or 5 is contained 3 times in 15, the numbers are all abstractly considered. On the other hand, a number is said to be concretely considered when it is considered in connexion with some particular objects, or particular units of quantity, which it counts or characterizes numerically. For instance, when we speak of 7 apples, 5 horses, 3 ounces, or 8 yards, the numbers mentioned are all concretely considered. Numbers abstractly considered are often called abstract numbers, while numbers concretely considered are, on the other hand, often called concrete numbers. It ought not, however, to be imagined that there are really two different kinds of numbers, abstract and concrete; the number in either case is purely a number, and not one kind of number out of more kinds than one. The whole expression five horses means a certain group of horses—a group characterized by the number five-and the five in that expression is purely a number. Also the whole expression eight yards is not a number, but is a quantity of length, or is the expression for a quantity of length. The 8 in that expression is purely a number; and if called a concrete number, the concreteness is merely in our employment of it as connected with the yard unit, so that the two come to denote jointly a quantity of length.

*The practice, which is rather common, of calling such an expression as £26-15-11 a number is objectionable, and can only be supported by some rather strained views of the nature of the expression; as, for instance, y saying that the 15 in it means 15 times 12 of the penny un ts, and that 26 means 26 times 20 times 12 of the penny units, and that so the whole expression denotes in a complicated way a certain number of the penny units. The simple state of the case is that the expression denotes 26 of the pound units of money, 15 of the shilling units, and 11 of the penny units; and so consists of three distinct numbers.

It is important now to observe that fractions in arithmetic may, like numbers properly so called, be considered either abstractly or concretely. If we speak of of an inch, or of a gallon, or of a yard, the numerical expression §, occurring in each of these three entire expressions, is an arithmetical fraction, applicable alike to the inch unit, the yard unit, or any other unit of any other kind of thing measurable in quantity. We could speak just as well of § of a pound of matter, or of a certain unit of power in steam-engines, water-mills, &c., called a horse-power. Thus the fraction may be spoken of and dealt with as disassociated from any particular kind of unit, but left applicable to units of quantity in general; and in this way it is abstractly considered. On the other hand, the whole expression five eighths of an inch means a quantity of length, and that length may be called a fraction of an inch. That quantity of length may be exhibited to the eye, or submitted to the touch, on an ordinary foot-rule, or by a little piece of wire cut of that length. The piece of wire would show and preserve the length intended, without any reference to the inch unit being required at all; but that same length, when specified by a numerical fraction conjointly with the inch unit, in being named of an inch, comes to be called a fraction of an inch. In this case the arithmetical fraction, being used conjointly with the inch unit to specify a quantity of length, may be said to be concretely considered; and corresponding statements might be made in respect to § of a gallon, of a pound of matter, or any unit of any other kind of thing measurable in quantity.

of

Throughout the foregoing explanations the word quantity has been applied in its proper sense, in which it is quite distinct from number, and signifies muchness, or signifies what is contemplated under the question how much? In this proper sense a great quantity of anything would mean much of that thing, while a great number of any things would mean many of those things. The word "quantity," however, is often extended in its signification by being taken to serve indiscriminately as a designation both for number and for quantity properly so called. This occasional extension of meaning of the term is subject to the great objection of tending to ambiguity of expression and to confusion of ideas, but still it is so frequent in the customary language of arithmetic, and more especially in that of algebra, that it ought to be known and understood. Indeed, the usage is so completely established in algebra that it can scarcely now be entirely discontinued; but in all cases of its employment care should be taken to have the intended meaning sufficiently shown by the writer and properly understood by the reader. In respect to the correct uses of the words "quantity" and "number," it may be remarked that we cannot properly speak of a quantity of water as equal to a quantity of length or to a quantity of time, but that we can properly speak of equality between the number expressing a quantity of water, and the number expressing a quantity of length, and the number expressing a quantity or period of time.