sense means Knowledge reduced to a system; that is, arranged in a regular order so as to be conveniently taught, easily remembered, and readily applied.

The practical uses of any science or branch of knowledge are undoubtedly of the highest importance; and there is hardly any man who may not gain some positive advantage in his worldly wealth and comforts, by increasing his stock of information. But there is also a pleasure in seeing the uses to which knowledge may be applied, wholly independent of the share we ourselves may have in those practical bencfits. It is pleasing to examine the nature of a new instrument, or the habits of an unknown animal, without considering whether they may be of use to ourselves or to any body. It is another gratification to extend our inquiries, and find that the instrument or animal is useful to man, even although we have no chance ourselves of ever benefitting by the information; as, to find that the natives of some distant country employ the animal in travelling;-nay, though we have no desire of benefitting by the knowledge; as, for example, to find that the instrument is useful in performing some dangerous surgical operation. The mere gratification of curiosity; the knowing more today than we knew yesterday; the understanding what before seemed obscure and puzzling; the contemplation of general truths, and the comparing together of different things, is an agreeable occupation of the mind; and, beside the present enjoyment, elevates the faculties above low pursuits, purifies and refines the passions, and helps our reason to assuage their violence.

It is very true, that the fundamental lessons of philosophy may to many at first sight wear a forbidding aspect. because to comprehend them requires an effort of the mind somewhat, though certainly not much, greater than is wanted for understanding more ordinary matters; and the most important branches of philosophy, those which are of the most general application, are for that very reason the less easily followed, and the less entertaining when apprehended, presenting as they do few particulars and individua 1 objects to the mind. In discoursing of them, moreover, n

figures will be at present used to assist the imagination; the appeal is made to reason, without help from the senses. But be not therefore prejudiced against the doctrine, that the pleasure of learning the truths which philosophy unfolds is truly above all price. Lend but a patient attention to the principles explained, and giving us credit for stating nothing which has not some practical use belonging to it, or some important doctrine connected with it, you will soon perceive the value of the lessons you are learning, and begin to interest yourselves in comprehending and recollecting them; you will find that you have actually learnt something of science, while merely engaged in seeing what its end and purpose is; you will be enabled to calculate for yourselves, how far it is worth the trouble of acquiring, by examining samples of it; you will, as it were, taste a little to try whether or not you relish it, and ought to seek after more; you will enable yourselves to go on, and enlarge your stock of it; and after having first mastered a very little, you will proceed so far as to look back with wonder at the distance you have reached beyond your earliest acquirements.

The Sciences may be divided into three great classes: those which relate to Number and Quantity, those which relate to Matter, and those which relate to Mind. The first are called the Mathematics, and teach the properties of numbers and of figures; the second are called Natural Philosophy, and teach the properties of the various bodies which we are acquainted with by means of our senses; the third are called Intellectual or Moral Philosophy, and teach the nature of the mind, of the existence of which we have the most perfect evidence in our own reflections; or, in other words, the moral nature of man, both as an individual and as a member of society. Connected with all the sciences, and subservient to them, though not one of their number, is History, or the record of facts relating to all kinds of knowledge.

I. The two great branches of the Mathematics, or the two mathematical sciences, are Arithmetic, the science of

number, from the Greek word signifying number, and Geometry, the science of figure, from the Greek words signifying measure of the earth,-land measuring having first turned men's attention to it.

When I say that 2 and 2 make 4, I state an arithmetical proposition, very simple indeed, but connected with many others of a more difficult and complicated kind. Thus, it is another proposition, somewhat less simple, but still very obvious, that 5 multiplied by 10, and divided by 2 is equal to, or makes the same number with, 100 divided by 4both results being equal to 25. So, to find how many farthings there are in £1000, and how many minutes in a year, are questions of arithmetic which we learn to work by being taught the principles of the science one after another, or, as they are commonly called, the rules of addition, subtraction, multiplication, and division. Arithmetic may be said to be the most simple, though among the most useful of the sciences; but it teaches only the properties of particular and known numbers, and it only enables us to add, subtract, multiply, and divide those numbers. But suppose we wish to add, subtract, multiply, or divide numbers which we have not yet ascertained, and in all respects to deal with them as if they were known, for the purpose of arriving at certain conclusions respecting them, and among other things, of discovering what they are; or, suppose we would examine properties belonging to all numbers; this must be performed by a peculiar kind of arithmetic, called universal arithmetic, or Algebra.* The common arithmetic, you will presently perceive, carries the seeds of this most important science in its bosom. Thus, suppose we inquire what is the number which multiplied by 5 makes 10? this is found if we divide 10 by 5-it is 2; but suppose that, before finding this number 2, and before knowing what it is, we would add it, whatever it may turn out, to some other number; this can only be done by putting some mark, such as a letter of the alphabet, to stand

.

Algebra, from the Arabic words signifying the reduction of fractions; the Arabs having brought the knowledge of it into Europe.

for the unknown number, and adding that letter as if it were a known number. Thus, suppose we want to find two numbers, which, added together, make 9, and multiplied by one another make 20. There are many, which added together, make 9; as 1 and 8; 2 and 7; 3 and 6; and so on. We have, therefore, occasion to use the second condition, that multiplied by one another they should make 20, and to work upon this condition before we have discovered the particular numbers. We must, therefore, suppose the numbers to be found, and put letters for them, and by reasoning upon those letters, according to both the two conditions of adding and multiplying, we find what they must each of them be in numbers, in order to fulfil or answer the conditions. Algebra teaches the rules for conducting this reasoning, and obtaining this result successfully; and by means of it we are enabled to find out numbers which are unknown, and of which we only know that they stand in certain relations to known numbers, or to one another. The instance now taken is an easy one; and you could, by considering the question a little, answer it readily enough; that is, by trying different numbers, and seeing which suited the conditions; for you plainly see that 5 and 4 are the two numbers sought; but you see this by no certain or general rule applicable to all cases, and therefore you never could work more difficult questions in the same way; and even questions of a moderate degree of difficult would take an endless number of trials or guesses to answer. Thus, if a ship, say a smuggler, is sailing at the rate of 8 miles an hour, and a revenue cutter, sailing at the rate of 10 miles an hour, descries her 18 miles off, and gives chase, and you want to know in what time the smuggler will be overtaken, and how many miles she will have sailed before being overtaken; this, which is one of the simplest questions in algebra, would take you a long time, almost as long as the chase, to corne at by mere trial and guessing (the chase would be 9 hours, and the smuggler would sail 72 miles :) and questions only a little more difficult than this, never could be answered by any number of guesses; yet questions infinitely more difficult can easily

be solved by the rules of algebra. In like manner, by arithmetic you can tell the properties of particular numbers; as, for instance, that the number 348 is divided by 3 exactly, so as to leave nothing over: but algebra teaches us that it is only one of an infinite variety of numbers, all divisible by 3, and any one of which you can tell the moment you see it; for they all have the remarkable property, that if you add together the figures they consist of, the sum total is divisible by 3. You can easily perceive this in any one case, as in the number mentioned, for 3 added to 4 and that to 8 make 15, which is plainly divisible by 3; and if you divide 348 by 3, you find the quotient to be 116, and nothing over. But this does not at all prove that any other number, the sum of whose figures is divisible by 3, will itself also be found divisible by 3, as 741; for you must actually perform the division here, and in every other case, before you can know that it leaves nothing over. Algebra, on the contrary, both enables you to discover such general properties, and to prove them in all their generality.*

By means of this science, and its various applications, the most extraordinary calculations may be performed. We shall give, as an example, the method of Logarithms, which proceeds upon this principle. Take a set of numbers going on by equal differences; that is to say, the third being as much greater than the second, as the second is greater than the first; thus, 1, 2, 3, 4, 5, 6, and so on, in which the common difference is 1; then take another set of numbers, such that each is equal to twice or three times the one before it, or any number of times the one before it; thus, 2, 4, 8, 16, 32, 64, 128; write this second set

* Another class of numbers divisible by 3 is discovered in like manner by algebra. Every number of 3 places, the figures (or digits) composing which are in arithmetical progression, (or rise above each other by equal differences) is divisible by 3: as, 123, 789, 357, 159, and so on. The same is true of numbers of any amount of places, provided they are composed of 3, 6, 9, &c. numbers rising above each other by equal differences, as 289, 299, 309, or 148, 214, 280, 346, or

307142085345648276198756

ich number of 24 places is divisible by 3, being composed of 6 numin a series whose common difference is 1137.