« ПредыдущаяПродолжить »
trace the science of numbers, from its early germs, till it acquired the strength and expansion of full maturity.'
Mr Leslie, accordingly, commences his history with the first rude attempts made by savage nations in the art of numeration. He leads us back to the infancy of society, when men have few ideas that are not excited by external impressions, and when they are consequently incapable of forming the very abstract idea of number. And he instances the savage fisherman or huntsman, eager to count over the produce of his exertions, and the bar barian leader, to marshal his followers and reckon up the slain, to shew that, from the wants and the wishes of such men, the feeble beginnings of science must have flowed.
He informs us that the first advance that would be made was to express number by palpable symbols. "If the numbers were small, they could easily be represented 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
After this first step had been made in the classing of numbers," it seemed easy and natural to break down each of these rows into similar parcels, and thus carry forward the successive subdivision till it stopped." The number of rows would probably at first be two; and, therefore, to represent, for instance, twenty-three similar objects, twenty-three of the smallest counters might be arranged into two rows, containing each eleven counters, and one over. But one of fourth, to place a single counter these rows would exon the fifth. By this binary arrange: press both, if counment, only four counters and one ters were taken of a double size; and, blank are sufficient to indicate the therefore, the two ows would be ex- number twenty-three; for the counpressed by other two, of five each, ters on the ascending bars, signifying and one over. By again doubling the two, four, eight, &c. the given numsize of the counters, ber is divided into sixteen, four, two, there would now be two and one. rows of two each, and one over. These two pairs might then be denoted by one pair of twice the size. And, lastly, this one pair might be expressed by a single counter.
"Hence the number twenty-three, as
Similar to this is the method of reckoning by threes, fours, &c. "The series of such natural emblems, however, is very limited, and would soon confine the range of decomposition. To obtain a greater extent, it was necessary to proceed by a swifter analysis; to distribute the counters, for instance, into ten or twenty rows, and to make pebbles, shells, or other marks, having their size only doubled perhaps, or tripled, to represent values increased ten or twenty fold." This would naturally" lead at last to a most important step in the ascent. Instead of distinguishing the different orders of counters by their magnitude, they might be made to derive an artificial value from their rank alone. It would be sufficient, for that purpose, to employ marks all of the same kind, only disposed on a graduating series of vertical bars or columns. The augmented value which these marks acquire, in rising through the successive bars, would evidently be quite arbitrary, depending, in every case, on a key to be fixed by convention."
Thus, if the values were to be doubled on each successive bar, instead of placing twenty-three counters on the first bar, it would be the same thing to place only one on the first, and eleven on the second; instead of cleven on the second, to place one on the second, and five on the third; instead of five on the third, to place one on the third, and two on the fourth; and, lastly, instead of two on the
If the values of the counters were now to be tripled, quadrupled, &c. on each successive bar, the numeration would be more rapid. Thus, if the number were expressed on the denury scale, as in the system of numeration
now adopted, twenty-three would be vast mound of earth, or a huge block
of stone, was the only memorial of any great event. But, after the simpler arts came to be known, efforts were made to transmit to posterity the representations of the objects themselves." "At this epoch of inprovement, the first attempts to represent numerals would be made. Instead of repeating the same objects, it was an obvious contrivance to annex to the mere individual the simpler marks of such repetition. Those marks would, of necessity, be suited to the nature of the materials on which they were inscribed, and the quality of the instruments employed to trace them."
represented by three counters on the bar of units, and two on the bar of tens.
On the same principle, any fraction may be represented on descending bars. Thus, employing the binary scale, "thirteen-sixteenths on the first bar, or the bar of units, are only equivalent to twenty-six such parts, or one counter and ten-sixteenths placed on the descending bar. This excess a gain corresponds to twenty carried to the third bar, making one counter and four-sixteenths. But these four-sixteenths, by successive duplication, pass over the fourth bar, | and leave a whole counter on the fifth." And thus the fraction consists of one-half, one-fourth, and one
This fraction, expressed on the ternary scale, affords an example of a circulating series. Thus, thirteen-sixteenths on the first bar are equivalent to thirty-nine, or two counters and seven-sixteenths on the second. This excess is equivalent to twenty-one, or one counter and five-sixteenths on the third. And this latter excess, passing over the fourth bar, leaves for
ty-five, or two counters, and thirteensixteenths on the fifth. But, since this excess is the same as the fraction on the first bar, the counters on the second, third, and fourth bars will be continually repeated in their order; and, therefore, the fraction thirteensixteenths is equivalent to the circulating series two-thirds, one-ninth, two-eighty-firsts, one two hundred and forty-third, &c.
The formation of such a series, says our author, "affords a fine illustration of that secret concatenation which binds the succession of physical events, and determines the various lengthened cycles of the returning seasons,a principle which the ancient stoics, and some other philosophers, have boldly extended to the moral world:" "Alter erit tum Tiphys, et altera quæ ve
The method of the Romans may be traced from this very simple origin. Having assumed a single line to represent one, or unity of the first class, they repeated the same character to denote two, three, four, and so on to ten. Representing ten, or unity of the second class, by two lines X crossing one another; they repeated also this character to denote twenty, thirty, forty, and so on to a hundred. And, in the same manner, representing a hundred, or unity of the third class, by [or C, a combination of three lines and a thousand, or unity of the fourth class, by M or CIO, a combination of four lines; they were able, by repeating and combining these characters, to represent any number from unity to ten thousand.
Delectos heroas; erunt etiam altera bella,
We must now pass with our author from palpable to figurate numeration. "In the early periods of society, a
To abridge the tedious repetitions of the same character, the Romans denoted five of each class by a part of the character for the next higher. They represented five by V, the upper half of X, (ten ;) fifty by L, a part of, the angular character for hundred; and five hundred by Iɔ, or D, a part of CID, the character for thousand. These repetitions they yet still farther abridged, by placing a numeral of less value before one of greater, to indicate the difference of their values. To represent the higher numbers, they conceived the value of I to be increased ten, a hundred, a thousand, &c. times, by simply repeating the character ; and the value of CIO, by repeating the character on each side of the I. And they placed a horizontal line over a numeral, to indicate that the value was increased a thousand times.
But, in the improved system of the
Greeks, the first nine numbers were represented by nine different characters; ten, twenty, and other collections of tens, by nine others; and one hundred, two hundred, and so on to a thousand, by nine others. Eight of the characters in each class were supplied from their alphabet, and the remaining three, by adding S to the first class to denote six, 4to the second to denote 90, and to the third to denote 900.
"In this manner," says our author, "the Greek notation proceeded directly as far as nine hundred and ninety-nine; but, by subscribing an ióta, or short dash, under any character, its value was augmented a thousand fold; or by writing the initial letter of myriad, the effect was increased still ten times more. With the help of punctuated letters, therefore, it reached to ten thousand, comprising four terms of our ordinary scale; and by means of the subscribed M, it was carried over another si milar period, or fitted to express a hundred millions. But the penetrating genius of Archimedes quickly discerned the powers, and unfolded the properties of such progressions. He took the square of the limit of the common numeral system, or a hundred millions, being ten thousand times ten thousand, for the index of a new scale of arrangement, which therefore advanced by strides eight times faster than the simple denary notation. This comprehensive series he proposed to carry as far as eight periods, which would hence correspond to a number expressed in our mode by sixtyfour digits."
After noticing the labours of Apolonius and Ptolemy, in improving and extending the numeration of the Greeks, our author thus introduces the system now employed.
their mental resources, overlooked the ad
"But those masters of science, rich in vantages resulting from a simpler mode of arrangement. They had only to ascend more slowly, and proceed by tens instead of periods of myriads; that is, to retain as numerals no more than the first set of their alphabetical characters, which were already employed with a point or short dash subscribed, to denote thousands. This might seem an easy step in the progress of invention, but the current of ideas had already flowed beyond it. Nor during the ebbing tide that preceded the fatal extinction of science among the Greeks, was any farther simplification effected, which would have shed a pale ray over the evening of that philosophy which was again destined to emerge from the thickest darkness, and relume the world. For the knowledge of our system of elementary numerals, which may be justly styled the Alphabet of Arithmetic,
we are indebted to a people extremely inferior to those instructors of mankind, in genius, acuteness, and general energy of character. Whether the Hindus lighted on that happy contrivance themselves, or derived it from their communication with
the natives of Upper Asia, there is yet no sufficient evidence to decide. however, to have become acquainted with They seem, it nearly two thousand years ago, and to have thenceforth commonly employed that mode of notation. From the Hindus again, their Arabian conquerors appear, areceived an improvement at once so simple bout the ninth century of our æra, to have and important. These industrious cultivators of science afterwards imparted the Moors, who still occupied the finest porvaluable present to their countrymen the tion of Spain. From this centre, it was gradually communicated over Europe. The earliest traces of the numeral characters among the Christians, may be referred to the end of the thirteenth century. They acks and astronomical tables; but their were at first introduced only into almangreat convenience soon brought them inte more general use."
We regret that our limits preclude us from following our author through the operations of palpable and figurate arithmetic; but we refer with great satisfaction to the work itself, for and many very interesting historical many refined artifices in calculation, illustrations.
Before concluding our imperfect account of this volume, there is one remarkable peculiarity which deserves to be noticed. Our author seems de
cidedly of opinion, that in language as in music, the ear must be satisfied; and that, as it is variety of tone that gives the relish to music, "variety, gives it all its flavour." The followalso, is the very spice of speech, that ing sentence may be taken as an instance of what we mean.
"A collection 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."
We do not pretend to justify, in every instance in which it occurs, the practice of our author in thus varying his mode of expression; yet, upon the whole, we see no great objection to his phraseology; and we think the severest of his readers ought at least to regard it with candour, as a laudable attempt to extend the language appropriated to the most perfect of the sciences.
Thy father sighed 'twas his to know,
To make the life an autumn leaf!
And fixed her eyes upon the spot. How soon is human bliss destroyed!-Life is a change of sun and shade, An undulating era, made Of pleasures passingly enjoyed, And hopes that blossom but to fade. An year hath circled o'er since then; The hopeful promise of a spring, The ripening summer forth did bring; The mellow autumn passed; again The winter spread his snowy wing.
The year revives; descending Night
Brood on the turf that mantles thee. "Tis thine with Solitude to dwell,
And silence, 'neath no storied urn; The dew may fall, the sun may burn, Upon thy little daisied cell,
At weeping eve, and day's return. No dark green cypress boughs are given, Above thy tomb to frown away, All undecaying, 'mid decay; "Tis courted by the breath of heaven, And open to the eye of day. And so it ought, sweet Innocent,
For Nature claimed thee as her own; A small, white, unelaborate stone, With Evening's dewy tears besprent, Marks out thy dwelling-place alone. M.
That find my soul estranged from thee! The past is but a pictured leaf,
Whereon, in glory, is displayed
Within thy smile, enchanting maid!
Thine image left my lonely breast ?-
A dark and fathomless abyss; Without a blossom, in the blight Of all that ever offered bliss! Forget thee!-ah beloved one,
It hath not been-it may not beThe sands of time have yet to run, That find my soul estranged from thee!