This is a reproduction of a library book that was digitized by Google as part of an ongoing effort to preserve the information in books and make it universally accessible. http://books.google.com 35 1225 42875 85 722516141251 36|1296 46656 86 7396 636056' 1369| 50653 1 87 7569 658503| 38 1444 54872 88 7744 681472 39 1521 59319 89 7921 704969] 4016001 64000 90 3100 729000 41 1681 68921 9l|8281 753571 4211 64' 74088l 92|8464 778688 43 1849 7950 8649 ;0435 44! 1936 85184 8836 830584 4512025 91125 95 9025 857375 46 21161 973361 96 9216 884756 472209103823 97 9409 912673 48 2301 110592 98 9604 941192 49 2401 117649 99 980 970299 50 2500 1 -J 5000 THE PHILOSOPHY OF ARITHMETIC; EXHIBITING A PROGRESSIVE VIEW or THE THEORY AND PRACTICE OF CALCULATION, WITH AN ENLARGED TABLE OF THE PRODUCTS OF NUMBERS- UNDER ONE HUNDRED. BY JOHN LESLIE, F. R. S. E. PROFESSOR OF MATHEMATICS IN THE UNIVERSITY. OF EDINBURGH. EDINBURGH: Printed by Abernethy % Walker, FOR ARCHIBALD CONSTABLE AND COMPANY, EDINBURGH ; AND LONGMAN, HURST, REES, ORME, AND BROWN LONDON. 1817- QA PREFACE. I now discharge my promise, by the publication of a volume, in which Arithmetic is deduced from its principles, and treated as a branch of liberal education. The object proposed was not merely to teach the rules of calculation, but to train the young student to the invaluable habit of close and patient investigation. I have therefore pre ferred the analytical mode of advancing, and have pursued a route entirely different from 1 hat which is followed by the common treatises of arithmetic. In seeking to unfold the natural progress of dis covery, I have traced the science of numbers, through the succession of ages, from its early germs, till it acquired the strength and expansion of full maturity. This species of history, combin ing sohd instruction with curious details, cannot fail to engage the attention of inquisitive readers. If the execution of this little work, which is the result of very considerable research, should at all correspond with the importance of its design, it may supply a capital defect in our systems of pu- 974 IV PREFACE. blic instruction. A great portion of the materials which compose it has already been given to the world, through the wide circulation of the Supple ment to the Encyclopaedia Britannica ; but the distinct and improved form in which it now ap pears is alone suited for general use. Some per sons will perhaps complain that it takes a wider scope than might answer all the common objects of tuition. But there is yet no royal road to knowledge, and whatever is acquired without ef fort becomes quickly effaced from the memory. Nothing indeed can be more fallacious than to ex pect any solid or lasting advantage from the sub stitution of a concise and mechanical procedure. The time required for the study of this treatise could scarcely be more beneficially employed, since it will not only rivet in the mind of the pupil the theory and enlarged practice of calculation, but invigorate, by proper exercise, his reasoning faculties, and consequently prepare him either for entering the labyrinths of business, or engaging in the higher pursuits of science. College op Edinburgh,! 15th October 1817. J INTRODUCTION. X he idea of number, though not the most easily ac quired, remounts to the earliest epochs of society, and must be nearly coeval with the formation of language. The very savage, who draws from the practice of fishing or hunt ing a precarious support for himself and family, is eager, on his return home, to count over the produce of his toil some exertions. But the leader of a troop is obliged to carry farther his skill in numeration. He prepares to at tack a rival tribe, by marshalling his followers ; and, af ter the bloody conflict is over, he reckons up the slain, and marks his unhappy and devoted captives. If the numbers were small, they could easily be represented by very portable emblems, by round pebbles, by dwarf-shells, by fine nuts, by hard grains, by small beans, or by knots tied on a string. But to express the larger numbers, it became necessary, for the sake of distinctness, to place those little objects or counters in regular rows, which the eye could comprehend at a single glance ; as, in the actual telling of money, it would soon have become customary to dispose the rude counters, in two, three, four, or more B INTRODUCTION. ranks, according as circumstances might suggest. The at tention of the reckoner would then be less distracted, rest ing chiefly on the number of marks presented by each se parate TOW. Language insensibly moulds itself to our wants. But it was impossible to furnish a name for each particular num ber : No invention could supply such a multitude of words as would be required, and no memory could ever retain them. The only practical mode of proceeding, was to have recourse, as on other occasions, to the powers of classifi cation. By conceiving the individuals of a mass to be dis tributed into successive ranks and divisions, a few compo nent terms might be made sufficient to express the whole. We may discern around us traces of the progress of nu meration, through all its gradations. The earliest and simplest mode of reckoning was bf pairs, arising naturally from the circumstance of both hands being employed in it, for the sake of expedition. It is now familiar among sportsmen, who use the names of brace and couple, words that signify pairing or yoking.— To count by threes was another step, though not practised to an equal extent. It has been preserved, however, by the same class of men, under the term leash, meaning the strings by which three dogs and no more can be held at once in the hand. — The numbering by fours, has had a more extensive appli cation : It was evidently suggested by the custom of taking, in the rapid tale of objects, a pair in each hand. Our fishermen, who generally count in this way, call every double pair of herrings, for instance, a throw or cast and the term warp, which, from its German origin, has exact ly the same import, is employed to denote four, in various articles of trade. INTRODUCTION. 3 These simple arrangements would, on their first appli cation, carry the power of reckoning but a very little way. To express larger numbers, it became necessary to renew the process of classification j and the ordinary steps by which language ascends from particular to general objects, might point out the right path of proceeding. A collec tion of individuals forms a species ; a cluster of species makes a genus ; a bundle of genera composes an order ; a group of orders constitutes a class ; and an aggregation of classes may complete a kingdom. Such is the method indispensably required in framing the successive distribu tion of the almost unbounded subjects of Natural History. In following out the classification of numbers, it seemed easy .and natural, after the first step had been made, to re peat the same procedure. If a heap of pebbles were dis- * posed in certain rows, it would evidently facilitate their enumeration, to break down each of those rows into simi lar parcels, and thus carry forward the successive subdivi sion till it stopped. The heap, so analysed by a series of par titions, might then be expressed with a very few low num bers, capable of being distinctly retained. The particular system adopted for this decomposition would soon become clothed in terms borrowed from the vernacular idiom. Let us endeavour to trace the steps by which a child or a savage, prompted by native curiosity, would proceed in classing, for instance, twenty-three similar objects. — 1. He might be conceived to arrange them by succes sive pairs. Selecting twenty-three of the smallest shells or grains he could find, he might dis pose these in two rows, containing each eleven counters, and one over. Having thus reduced the number to eleven, he might sub 4. INTRODUCTION. divide this again, by representing only one of the rows, with shells twice as large as before: He would consequently obtain two rows of • • • • • ^ Jive each, with an excess of one. Instead of these shells, were he to employ shells of a double size, it would be sufficient to denote one of the rows, 9 m or to dispose it into two rows. These rows con- • • • tain each only two counters, with one remaining. Again, by adopting counters of a double size, the last row % might be represented by one pair, each contain- 9 ing only two marks exactly. These again could be deno ted by a single counter, of twice the former di- ^ mensions. — Hence the number twenty-three, as decomposed by repeated pairing, would be denoted by one counter of the fifth order, one of the third, one ^ of the second, and another of the first. # • 2. Again, suppose a person should attempt to represent the same number, by triple rows of shells or counters. He would first have seven of the smallest shells in each row, with an excess of two. Then, expressing one of the rows by shells of three times the size or value, • • it would be again resolved into three * ^ rows, each containing two counters, with an excess of one. But these rows might be repre- % sented by two counters of triple size ; % and here the decomposition stops. The number twen ty-three is thus expressed, on the sys tem of triplication, by two counters of ^ ^ the third order, one of the second, and * two of the first.