POPULAR MATHEMATICS.

SECTION I.

GENERAL REMARKS, AND DEFINITIONS.

THE Common opinions of mankind upon a subject are frequently at very remarkable variance with the nature of that subject; and this variation is not perhaps more striking in any one case than in that of Mathematics. Those who have never studied any portion of mathematical science, however acute they may naturally be, and however well informed upon other, and in themselves more difficult subjects, generally, if not invariably, turn away from every mathematical expression, as if it were an adder in their path; and even they who, to use the homely but most appropriate expression, have "gone through" that which is called mathematics at the common schools, shake their heads at the subject, with a silent expression of, "These matters are beyond our depth." The conduct of such parties puts one very much in mind of that of the porter in a northern University. This porter was a very "whale" of books, and one of the professors, whose particular attention he claimed, found the supplying of his appetite from the University library no easy task. At length he tried him with " Euclid's Elements

B

2

EUCLID AND PTOLEMY.

of Geometry," to see how far sheer appetite would be able to digest that. The porter came not for an exchange till after two weeks had elapsed; and at last he came, somewhat crestfallen, saying, "Docter, I hae read a' the wirds, anʼ leukit at a' the pikters, but it's the maist puzzleanimous beuk I hae seen, an' I dinna onderstand ae wird o't; sae ye'll jeust hae the gudeness to gie me a beuk that has nae As nor Bs in't."

It is probable that some part of this general dread of mathematics may have been occasioned by the reply of Euclid to Ptolemy Philadelphus, the Egyptian monarch. The king wished to know if there was not an easier method of learning geometry than that which was practised in the schools; and the mathematician bluntly, but somewhat ambiguously replied, "There is no royal road to geometry." Now, all that was meant by these words was, that geometry must be studied by man as man, and not as monarch; that it must be conquered by the mental exertions of the individual alone, and not by any subjects which he can command, or any armies that he can muster; so that, if we take it in its true meaning, the saying of Euclid is an express declaration, by one of whose judgment no one can doubt, that any man might be a geometer if he would bring his own mind to bear upon the subject; and that in this science, the civil and political distinctions of mankind go for nothing, for it is as open and as plain to the humblest peasant as to the proudest king.

Sixty-three generations of men, at the average allowance of one-third of a century for each, have been born and have died since this reply was given by the Alexandrian geometer; and during this long period, men of all ranks, from the monarch to the peasant, have studied and promoted geometry, and the other branches of mathematical science; but this reply has been brought forward as a sort of bar in the way, not of kingly

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 may 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