information are not before us, we merely have the abstract mental conceptions of them expressed by certain conventional symbols, the letters of the book, or the sounds of the words which are spoken; and it is because the mental conception comes home at once to our own minds, without any of that laborious examination in detail which objects of the senses require, that the knowledge is both so powerful and so permanent. In the real object we can see but one side at a time; and the story of that which we observe does not extend either way beyond the time of our actual observation, without drawing our attention from the reality. But the mental embodyment is transparent: we can see all sides of it at once; we can view it in all its succession of states; and we can bring it forward and study it whenever we please.

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Every one must see the advantage of thus being able to bring the mind to bear upon the whole of a subject, in all its parts, in all their connexions and relations, and in succession of time; because we are thereby enabled to see the end from the beginning." In the forming of any plan we can not only see whether the ultimate object is attainable, but we can see the shortest way to it, press into our service all that can promote its success, and remove all that would occasion hindrance or failure; whereas those who can take no such mental view, but must at every step "wait till they see," are constantly bungling and blundering, and really have more trouble in correcting their own errors than in all the rest of the business of life. We do not mean to say that every one who has this capacity of forming skilful plans, either does form them or carry them into execution; but it is abundantly clear that the capacity must exist before the plans can be formed.

It is not to the projecting, or planning, of any one thing that this capacity applies; for it applies to every thing, from the

greatest action in which men can be engaged to the simplest project of the humblest individual; and it applies as well to the conduct of the execution as to the formation of the plan; so that it is in truth the grand rule of life. If it were possible to impart this capacity to every individual, or rather to train every individual to this habit, (for after all it is merely a habit, and a habit which calls for no extraordinary power of mind, but would adapt itself to the mind of every individual,) it would cost less labour in the learning and the observance than people must necessarily undergo in consequence of the want of it, and not only so, but the life of man would in point of efficiency be greatly lengthened. As things are managed at present, a large portion of the time of most people is occupied in returning from the wanderings, and rectifying the blunders of the rest of it; and it is perfectly evident that, if they could be spared this useless labour, they would have all the time it occupies as leisure, during which they might increase their gains, or improve their minds, or enjoy themselves, just as suited their fancies; and thus, while there would be an end of much needless labour and real suffering, there would be a corresponding increase of efficient labour and real enjoyment; for it must be remembered that the value of labour is not measured by time, but by productive power; and that the enjoyment of labour is never so sweet and so satisfactory as when we feel that we have earned it by doing our duty to ourselves and our country in the most perfect and effective manner.

This mental power, in the extended sense in which we have viewed it, cannot be said to be mathematics or mathematical; because it applies to all subjects, while the subjects of mathematics are limited. But the difference is one of subject rather han of principle, and the conduct of the mind in the cases alluded to is in strict accordance with its conduct in mathema- .

tical investigations. Indeed, there is so much of similarity, that if we take out those portions of any particular case which are contingent, all that remains is strictly mathematical. The contingent parts are the data obtained from without-the results of experience and of testimony; and though these, taken in the general sense, are not mathematical, yet that keen scrutiny into the truth or the falsehood of everything which a mathematical habit produces, is of great use in estimating the worth both of observation and of that which rests upon evidence. Thus we are to consider the strictly mathematical habit in the mind, which is the most valuable part of the science, to consist in arranging according to their proper relations all deductions which the mind itself is capable of drawing from self-evident principles, and in discovering truth and detecting error in every possible combination which can thus be formed. In the doing of this, there can be no half measures; for to what extent soever we may arrive in the combination, we must be able at every instant to look back to the very outset and make sure that every single step, whatever may be the number, has been taken rightly. Thus, for instance, if the subject of immediate inquiry is the distance of the sun, we must be able to see our way backward from this great distance to the measuring of a straight board with a foot rule, and how we have been able to find our way from so short a measure to one which is so vast. In like manner, if the question be to what height the attraction of the moon shall raise the tide of the ocean, we must be able to connect this with the weighing of a pound or an ounce by the means of a common balance. Generally, whenever we are to apply our mathematics to the ascertaining of anything which we wish to know, however great or however complicated, we must see our way not only from some operation which we ourselves could actually perform, but from some principle so simple and so clear

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USE OF MATHEMATICS.

that nobody who understood the words could refuse to give their assent to the truth.

These are great advantages-advantages which, in a mental point of view, we seek for in vain in any other department of human knowledge; and it will readily be admitted that if we are trained and habituated to this extent and transparency of mental vision upon a number of subjects, we will endeavour as a matter of course to exercise the same, as far as it may be practical, upon every subject. Then in addition to this there are the practical applications, which include all calculating, and weighing and measuring, and comparing, and estimating, and determining value of every kind. In a word, if we take mathematics from all the practical knowledge which we find of so much use to us in the occupation and business of life, we should leave nothing behind but uncertain guesses and conjectures, and could not by possibility be either a successful or civilised people. Thus we cannot, and whether we know it or not we do not, manage matters without the virtual aid of mathematics; and why should we not get the real aid? There are no doubt a great many technicalities, and mathematical writing has the appearance of being in a strange tongue. But this is a mistake; the difficulty is not so formidable as it seems, and the language is not only our language, but the language of all nations who will give themselves the trouble of learning that which every child learns first, namely, an alphabet.

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SECTION II.

SUBJECTS, OBJECTS, AND PRINCIPAL DIVISIONS OF MATHEMATICS.

QUANTITY is the subject of all mathematical investigations and proceedings, whether theoretical or practical, that is, whether relating to the discovery of general principles and relations, or to the application of these to particular cases. Therefore, before we can enter upon the science with any chance of success, it is necessary that we should clearly and perfectly understand what is meant by quantity.

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Quantity, from the Latin quantus, literally means as much as there is;" and it is easy to see that the words “many, large, great, long, quick," and an endless variety of others, may be used instead of the word "much," or that the word "much" may be retained, and the other word added to it, being at the same time changed to a noun. Thus, “as much of largeness as there is," and so in all cases. But we cannot thus turn the word "much" into a noun, and use any of the other words that satisfy the meaning of the sentence when alone as an adjective before it. Thus we can see that the word "much" is a more general one than any of the others, and can be applied to every kind of quantity, while the rest apply to particular kinds or particular modifications.

We could with equal propriety use the word "little,” which refers to quantity in the same general sense as "much" does, and the difference between them is a matter of relation and not of reality. This will readily be perceived when we consider that the very same quantity can and would be considered as much by one party and little by another. Thus five pounds in money would be much to a poor labourer for a week or even a

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