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power, but of intellectual ability, and the consequence has been that, even at this day, and in this country--the foundation of whose greatness has been mathematical science, the great body of the people know less of the principles of mathematics than of those of almost any other subject. And even that love of reading which has of late years been so generally diffused, and which
be made the instrument of so much good, has not embodied anything like a fair proportion of mathematical knowledge, neither have they who have gone about to cherish this love by the multiplication of books of small size and easy price, done anything like justice to the public in this respect. The mathematical tracts which they have produced are few in number, and as to their value, it is to be feared that it is still less.
With the cause of this deficiency we have no immediate concern : but the probability is, that it is found impossible to compile mathematical books—to take a little bit of one, and a little bit of another, and tack them together into an amusing miscellany, any page of which may be read with at least some sort of understanding, without reference to the rest. Or it may be that we possess no mathematicians but such as are professionally so; and thus, however able they may be in a professional point of view, they can treat the subject only in a professional manner, and would consider their labours deteriorated and themselves degraded, if they were to abate one iota of the technicality of the schools. Now we are very ready to acknowledge the full value of this technicality, and to admit that every apparent difficulty in mathematics, is essentially a simplification. We do this confidently ; because mathematics is, as we shall show by and by, the only portion of science which has hitherto stood, and must for ever stand, impregnable to the mere book-maker; and that no man can put a single pin to this fabric without putting the right one, and putting it in the
right place. But still, perfect and beautiful as is this technical structure, and proudly as it towers over the rest of human knowledge, as the noblest conquest and heritage of intellect, and frowning defiance and scorn against every species of imposture, it is too mighty for any but those who are to give themselves wholly up to it. At the same time, as it is the purest exercise of the mind, the real instrument of discernment, that in which the individual must be thrown wholly on his own strength, it is desirable that some portion at least should be accessible by every one who can read, and that this general portion should not be those insulated scraps of the applications which are useful to men in particular professions, but at least as much of the principles, as shall give a mathematical turn to the mind, which is but another name for precision and accuracy of thought.
It may seem paradoxical, but it is nevertheless true, that however ignorant we may be of the forms of mathematics, and how much soever we may regard the technical expressions of the different branches of mathematical science as puzzles or mysteries, we are all mathematicians in reality ; and the process by which we arrive at the precise and accurate knowledge of any one subject whatever, is really a mathematical process, whether we know it to be so or not. The only difference, indeed, between one who understands the principles of mathematics, and can apply those principles to the finding of results, and one who must get at the results the best way that he can, without any knowledge of the principles, is, that the first proceeds with ease and certainty, while the other proceeds with great labour, and is doubtful of the result when he has arrived at it. Mathematics, to use a homely comparison, may be compared to tools and the capacity of using them ; while the subjects upon which mathematics are exercised are the materials,
out of which that which is desired is to be formed by means of the tools ; so that a mathematician stands to a man who is no mathematician in the same relation as a clever workman well furnished with tools stands to a man who has no tool and no knowledge of the use of one ; and when we look at the accommodations of civilised men, and compare them with those of men at the bottom of the scale of savagism, we are able to judge of the difference between the man who possesses the instrument and knows how to use it and the man who is ignorant of both. The disparity is even greater than this; because mechanical operations, valuable though they be, are only one particular case, whereas mathematics reach every operation of the mind, give clearness to every thought, and regulate with. certainty every action.
One other cause of the ignorance in which mankind suffer themselves to remain of mathematics, may possibly be want of knowledge of what the term means; and this is rendered the more probable by the fact that, in the ordinary way of teaching the individual branches of mathematical science, such as arithmetic, or the elements of geometry, the student is sent to the details of the subject at once, and without any preliminary explanation of the use, or even the general nature of what he is called upon to do. The consequence is, that there is no goal before him, nothing to keep alive his hope, or rouse his mental ambition ; and so he drudges on like a slave, measuring his labour by the day, and his pleasure by the smallness of the quantity of the day's labour. Upon young minds especially this has a most baneful influence; as it not only destroys the possibility of progress in mathematics, which must either be a labour of the willing mind or no labour at all, but becomes a habit, which is transferred to and which destroys every other branch of education, and perverts and poisons every course of
future life. How much of this preliminary explanation should be given in any case must depend on the nature of that case itself, on the age, ability, and previous knowledge of the student ;
but in every case the danger is that it shall be toa little, and not that it shall be too much.
It is perhaps difficult to convey in writing even the simplest outline of what should be done in such cases, because it belongs to the province of colloquial instruction—that in which the instructor can lay hold of present and visible illustrations, and vary, and reiterate again and again, with a tediousness which no ordinary reader would tolerate in print, but which in practice is the only sure way of “trying for the vein ” which will .make the mine of instruction work easily, certainly, and profitably. The few sentences which follow must therefore be considered, not as furnishing what is to be done, but merely as giving a hint that, in order to insure success, something ought to be done.
MATHEMATICS (MáØnsis) contains in the name itself no bad general definition of the whole science, or rather the mode, so to express it. Thesis means a position, that which can be either true or false, but it leaves the mode of arriving at the truth or falsehood perfectly general, although it always does involve in it the notion that there is some sort of proof; and as the discovery of a falsehood is a truth, though the falsehood itself is not, every useful thesis may be considered as the statement of a truth; and the truth which amounts to a thesis must not be one which is perfectly apparent to every body without any proof or argument. Thus, if we were to say
It rains," to a person whom we met out of doors during a shower, the saying would be no thesis ; neither would it be a thesis to say “ The sun shines,” to one whom we met in the fields at noon on a cloudless day. In both these cases the party whom we
addressed would understand the fact just as well as ourselves, and therefore our observation would be an idle waste of words. But if we were to say “ The apparent motion of the sun westward is owing to the real rotation of the earth eastward,” it would be a thesis ; because it is a truth which cannot be arrived at by simple observation of the sun, but quite the reverse; and therefore, to a person ignorant of the motions of the earth, we would require to bring forward proofs before we could call upon him to believe it. Thus we may consider every truth which requires to be established by reasoning, by evidence, or in any way whatever, as a thesis ; and it is not, properly speaking, a thesis until the proof is given, for this simple reason, that no truth can be regarded as such until it is known to be so.
When no proof has been given, but there is still some probability that a position may be true, it is a proposition, or hypothesis, which means something which precedes, or is inferior to, a thesis ; and which requires to be proved before it can be elevated to that character. If the proof shall afterwards be obtained, the hypothesis takes its rank as a thesis, and becomes a portion of knowledge; but if the proof fail, the hypothesis falls to the ground as a vain and unsuccessful attempt.
The methods of proof employed for the establishment of different truths are so exceedingly numerous, that a list of them would be long, not very interesting, and out of place here; but still, in order to see clearly the nature and use of mathematical proofs, or which is the same thing, the mathematical modes of establishing truths and rejecting falsehoods, it is necessary to know something about the general divisions of proof. The simplest view which can be taken of this subject is that which divides the whole into three great classes-observation, testimony, and proof by reasoning. Observation is only another name for that of which we have the evidence of the senses;