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(so to speak) of the mathematical art, but we cannot be permanently deceived without our own consent—we can always test our result. This is the grand distinction—we can insure truth. If we consider any fact in history, and then compare it with the deductions of Geometry or Arithmetic, &c., the wide difference between mathematics and all other branches of human knowledge becomes strikingly evident. We cannot demonstrate the existence of Lycurgus and Solon, of Hannibal and Mahomet; nay, we can deny that such men ever lived, and we shall contradict no necessary truths of the human mind-there will be nothing absurd in the denial. But in denying a proposition of Euclid or a process of Arithmetic we shall controvert the judgments of our own consciousness, and fall into utter absurdity. The truths of mathematics, therefore, are absolute, while those of all other sciences are only probable. Our belief in history, for example, is wholly grounded on evidence; the facts, if such they be, are beyond our experience, and are not cognizable by our self-consciousness. In truth, the flimsy, groundless pature of many of our beliefs, is almost ludicrous. We are sometimes no whit more sapient than Smith, the weaver (when he avowed his belief in Jack Cade's assertion, that his father was a royal heir, who had been stolen away in infancy, and brought up as a bricklayer)--"Sir, he made a chimney in my father's house, and the bricks are alive at this day to testify it, therefore deny it not." Thus we long rested in a belief of the personality of Homer, chiefly because the “Iliad" and “ Odyssey ” were "alive to testify it;" in later days, critical acuteness has disturbed our faith, and reduced the existence of the great father of song to a mere topic of debate. So Romulus and Remus, and their successors, were acknowledged and believed in from the days of Livy to those of Niebuhr, when we were suddenly compelled to new model our ideas of Roman history, and to condemn our old facts as so many antiquated fables. Examples of this kind are numerous enough, even as regards what are termed the facts of history; but when we come to consider the deductions drawn from those facts, the uncertainty of our knowledge becomes startingly obvious. How many versions have we had of the character of Cromwell! Which, then, is the truth? was he indeed a hypocrite and a tyrant, or a model of piety and a chosen servant of freedom? Again; we have seen the claims of Napoleon and Wellington debated in these pages. How unlike the different portraits drawn! yet each is doubtless drawn in all faith and sincerity. Nor is this uncertainty—this mere feeling after truth, "if haply we may find it”-confined to history; it characterizes other departments alike. Theology and politics, science and social economy, each afford ten thousand topics for debate. The truth of one age is often the scorn of the next. How different is all this from mathematical truth! The grand imaginings of Plato have sunk and risen repeatedly, and their splendours have paled in the broad daylight of Christianity; the philosophy of Aristotle has wielded the sceptre of empire, and again dwindled away into comparative obscurity; the astronomy of Ptolemy has been exploded for centuries, and the life-long labours of the schoolmen are almost forgotten;- yet in our very schools the youthful aspirant of to-day toils at the demonstration of the fact that the squares formed on the two sides of a right-angled triangle which contain the right angle, are together equal to the square formed on the side opposite
right anglo-a faet ascertained and proved by Pythagoras 550 years before our
Sur's birth, or 2,400 years ago! How grand the thought, thus to contemplate a spark f eternal tire flashing through darkness of abore two thousand years! How great a testimony to the immutability of mathematical truth, that in the nineteenth century the university students of modern Europe should use the very book which Euclid wrote or compiled 300 years before the christian era! Does not each one whose aim is truth, and who seeks in self-culture an approximation to its attainment, sympathize with the joy these ancient sages must have felt, when, amid the gross darkness of paganism, and the contending theories of philosophy, they were enabled thus occasionally to grasp the real and the infinite, the absolute truth of DEMONSTRATION? Will not our readers resolve to share this joy, to toil for such a prize? To one and all of such students we commend the
immediate study of the principles of Arithmetic as the first step towards the sublimer = truths of the higher mathematics.
But not only do we coinmend this department of study from its mental grandeur and dignity, but also for its utility. The tendencies of our age are eminently scientific and calculative. Mathematics are the key to physical science. It is almost absurd to expect any one to have a knowledge of science until he brings to the study of it a good knowledge of mathematics. It is vain to read Compendiums and Introductions, Guides to Science,
Dialogues, &c. Such works are useful in their own way,—they may serve to give a bias - to the youthful mind towards science, or to throw some rays of light athwart the darkness
of the minds of those who either have not the time or the opportunity to study the wonders which surround them. But if any one would be something more than a smatterer,-if he would know the luxury of thinking for himself, of finding out the why of simple phenomena instead of being “content to be told,” of understanding instead of trusting,-if he would have an intelligent knowledge of the great unities and guiding principles of science, instead of being a sort of live catechism or animated glossary, then let him resolve to be a mathematician. Again, not only for science, but also for everyday life, a knowledge of mathematics is essential. How could the commerce of England have arisen to its colossal magnitude, if arithmetic had remained in the clumsy condition of former ages? Where would be our banking and stockbroking, our Exchanges and Money Markets, our Boards of Trade, &c., &c., if we were compelled to compute interest by means of the abacus of the ancients, or to calculate foreign exchanges with counters like our forefathers? The Bank of England is as dependent upon the arithmetic of its clerks, as upon the bullion in its cellars. In every grade of life this knowledge is more or less essential; the Chancellor of the Exchequer and the humblest day labourer are both compelled to calculate, in order to measure their wants by their means, and to make both ends meet." The wealthy man, who insures his life for thousands, and the poor member of a provident society, alike stake the happiness and comfort of themselves and their nearest friends on the correctness of arithmetical calculations.
Lastly, we commend this study on individual and personal grounds, as a means of mental discipline. As frequently studied, we fear that arithmetic at least is rather an injury than a benefit to the intellectual powers of the student. It is too often a system of rules and routine, fatal to our independence of mind, because directly opposed to that selfreliance and industry upon which all true independence of mind is founded. Well has Condillac remarked :
"Ce n'est point par la routine qu'on s'intruit, c'est par sa propre réflexion; et il est s essentiel de contracter l'habitude de se rendre raison de ce qu'on fait; cette
habitude s'acquiert plus facilement qu'on ne pense; et une fois acquise elle ne se perd plus."*
It is this power of reflection, of independent thought, which constitutes the true dignity of manhood. Until this is obtained, we are bondsmen in mind though free in body, mere mirrors reflecting the living objects which float around us. The value of the higher mathematics (especially geometry) in creating or calling forth all the energies of thought of which we are capable, and of training them in habits of severest reasoning, is too well attested by the example and advice of the most eminent men of all ages to need either enforcement or illustration at our hands. From Plato down to the Mills, Herschels, Broughams, and Sedgwicks of to-day, there is scarcely one dissentient voice; the annals of statesmanship, and the records of our law courts, all prove this branch of study to hold the first rank as a means of discipline. Amid the cares and anxieties of life, whether the fate of nations or the interests of individuals are concerned, the greatest blessing which can be possessed is the power of mental abstraction and concentration,—the calm mind and clear grasp of thought, which constitutes the idiosyncraey of a great mathematician. In every view, therefore, whether considered with reference to its own characteristics, its social and scientific uses, or its personal effects, this study has the highest claims on the self-cultivator.
Arithmetic, the branch of mathematics to which we now direct attention, must have been, to some extent, coeval with the human race; we cannot conceive the possibility of exchange and barter, or any other of the simplest incidents of social existence, unless we also bring in some rude idea of numeration. The veriest savage must have had some way of registering the number of his sheep or cattle, &c., even though he may have had no means of expressing numbers in language; thus he might represent his sheep symbolically by a heap of stones,-a stone for each sheep; adding to the heap and taking from it, just as his sheep increased or diminished. This undoubtedly was the origin of arithmetic. Calculation was the use of calculi (pebbles), a manual arithmetic. The ancient Romans kept count of time in this way. The “ Gentleman's Diary" of those days was a box or urn, into which the owner every evening dropped a lapillus (small stone),—a white one if the day had been a happy day, a black one if the opposite. So, once a year, the prætor went in great pomp to the temple of Jupiter Capitolinus, and drove a nail into the door of the temple, so as to keep a correct record of the years the city had been built. This manual arithmetic, though easy, inasmuch as it saved all thought, and reduced the matter to a mere mechanical art, must have been sadly cumbrous and tedious. The Roman schoolbog (Horace, Sat. I. vi. 73), with his “ loculos tabulamque” † hanging on his arm, must have found that arithmetic was no light matter. The same and ruder kinds of physical arithmetic were prevalent in our country. The accounts of the exchequer (seaccarium, table of chequers) were long kept by means of tallies or rods of seasoned willow or hazel, which were notched so as to represent the various sums of money. In ordinary life, counters were long used in most of the operations of arithmetic. Shakspere represents a clown in
• Self-culture is not to be obtained by routine, but by individual thought; it is essential to acquire the habit of giving a reason for everything we do. This habit is more easily acquired than persons generally suppose, and when once acquired, it is never lost again.
+ Calculating frame and box of pebbles.
the “ Winter's Tale,” utterly perplexed with a very simple calculation, and complaining, “I cannot do't without counters” (Act iv. sc. 2). Again, in “ Cymbeline," we find one of the characters thus praising his craft, and comforting his prisoner :
“Gaoler. A heavy reckoning for you, sir; but the comfort is, you shall be called to no more reckonings. '... Oh! the charity of a penny cord! it sums up thousands in a trice. .... Your neck, sir, is pen, book, and counters ; so the acquittance follows."
An attempt to revive this species of “palpable arithmetic" was made by the late Sir John Leslie, as conducive to the better understanding of the philosophy of arithmetic; but the remembrance of his treatise is passing away, and with all due deference to his brilliant talents and acquirements, we conceive that it would prove a failure with those who would not think, and a useless waste of time with those who are willing to study the principles of arithmetic. As a practical matter, then, we may pass by these forms of external arithmetic; as a subject of history and curiosity, the student will find much interesting information under the heads Abacus, Loculus, &c., in the various Encyclopædias, and the different theoretical and historical treatises on arithmetic.
We shall now briefly trace out the origin of oral and written arithmetic, and thereby endeavour to illustrate the nature of the science. Repetition, or multitude, is among the earliest ideas of childhood, and is the basis of number, the fundamental idea or subject matter of arithmetic. The passage, however, from the vague idea of multitude to the more definite one of number, is one of some degree of difficulty. “It requires an analysis of the individual* units of which a number is composed, which can only be effected by the comparison of different numbers.” Hence the child is easily confused in his attempts to number, and the savage is driven to mere vague analogies whenever a number transcedes his usual limits of account, pointing in helpless but significant perplexity to the sparkling bosts of stars above, or the myriad grains of sand below. Nor do we gain the precise idea of number, as the subject matter of arithmetical science, from this mode of analysis and comparison; we learn to distinguish three men from four men, but it is much in the same way as we distinguish a man from any other object, i.e., as differing in certain attributes. To obtain the philosophic idea of number requires the exercise of the power of abstraction, which is the creature and result of language. When we become accustomed to give to the ideas three-men and four-men a mental existence by means of representative words or terms, we soon perceive that such ideas are compound, -each possession in common the simple idea“ man;" it then becomes possible to separate the idea of three-ness and fourness (if I may be excused these words) from the idea “man,” and to regard " three" and “ four” as independent ideas separable and distinct from any objects with which they may be conjoined, -in other words, we arrive at the abstract conception of number. Here a difficulty would soon meet us. Each number must have a distinct name, and therefore the language of arithmetic would soon be in danger of becoming cumbrous and intolerably verbose. This, indeed, has proved the stumbling-block of arithmetic among all uncultivated nations.
* Hence number is frequently defined to be a collection of unities, and attempts are made to define this term. The two ideas, however, are among those primary conceptions which may be illustrated, but cannot be clearly defined, and all attempts to do so tend rather to confuse the mind; we take it, therefore, for granted that our readers will not expect us to attempt so ungrateful and useless a labour.
Aristotle tells us of a Thracian tribe who could count no higher than four, and La Condamine informs ns of a South American tribe, whose name for three is Poettarrarorincoarroac! As might be imagined, this euphonious polysyllable has proved an arithmetical Rubicon, which the unfortunate Indians found themselves utterly unable to pass. The difficulty among more ingenious nations has been more or less successfully overcome by means of different systems of classification. At the present day, as is well known, we classify by tens, or, in other words, use a decimal system of notation in our ordinary arithmetic. We have separate words to express the first ten numbers (one, two, three, fc. . . . ten), and then we manage to express the higher numbers by means of combining these simple names, so as to avoid the difficulty, confusion, and verbosity alluded to before. Thus we continue our numeration after ten as follows:
Eleven,* i. e., one-and-ten.
&c. On arriving at the end of the second series of tens, i. e., ten-tens, a new word is intro duced, hundred, by means of which we are enabled to extend our numerations to tenhundreds, or, as it is also called, a thousand, without any appreciable increase of complexity or clumsiness. In like manner, by the use of “thousand,” we are enabled to extend our numeration so as easily and briefly to express any number up to a thousand-thousands, which we again abbreviate and designate by a single word, million. Beyond this we have the terms billion, trillion, &c., but they are comparatively seldom used, being numbers too vast for us to grasp as actual conceptions, either in the abstract or concrete form.
To the thoughtful mind there is something very beautiful in the comprehensive simplicity of this system of notation. About a score syllables, by their various combinations, enabling us to frame a distinct, clear, and intelligible name for so many variations of numeral conception! Most striking is the lesson to be drawn from hence, as to the value of classification, the capabilities of language, and the influence of scientific principles! May it lead all, who have not hitherto done so, immediately to commence the study of arithmetic as a science not unworthy of the dignity of man. Let all resolve to take up the study in the spirit of the old Greek Platonist (Theon of Smyrna)—“Arithmetic should not be studied with gross and vulgar ideas, but in such manner as to attain to the knowledge of numbers, --not for the sake of dealing with merchants and tavern keepers (alone), but for the improvement of the mind, considering it as a path leading to knowledge and reality.” When we consider the order which reigns in the universe, and reflect on the fact that the regular recurrence of phenomena which constitutes order (and produces the impression of a designing Cause governing the universe it has framed) is subject to and expressible in arithmetical form, we can scarcely wonder at the enthusiasm of the Grecian
* According to recent philologists, leren is connected with the Latin decem.