clown, staggered at a simple multiplication, exclaims, "that he must try it with counters." 9. Although any number whatever might have been employed as the index or root of the arithmetical scale, yet the selection is not a matter of mere indifference. If the root were smaller, a less number of distinct characters would be sufficient to express any number; but these would require to be oftener repeated. Thus, the Binary scale only requires one written character expressive of number, whereas the Denary requires nine. To express the same number, however, on these two scales, more characters will be required by the Binary than by the Denary, as will be evident by expressing any number, suppose eight hundred and forty-five, on both. By the Binary it will stand thus: 1101001101; and by the Denary thus: 845. The Denary scale, which was generally used by the ancients, and now universally by the moderns, is perhaps better fitted for arithmetical computations than any other of those scales which have been mentioned, but is not the best that might have been adopted. The index or root ten of the Denary scale has only two aliquot parts, which are two and five; in other words, ten is only divisible by the numbers two and five, without any remainder. Had the Duodenary scale been adopted, whose index, twelve, is divisible by more numbers than that of the Denary, viz. by two, three, four, or six, a number expressed on the Duodenary scale would have been divisible by more divisors than if it had been expressed on the Denary; and hence fewer fractions would have occurred from the division of numbers expressed on the former scale than do from those expressed on the latter. It is a curious fact, that the celebrated Charles XII. of Sweden, a very short time before his death, seriously deliberated on a scheme of introducing this system of notation into his dominions. He seems also, at another time, to have had a strong desire of changing the index of the arithmetical scale from ten to sixtyfour, because 64 is both a square and a cube number, and when continually divided by 2, is at last reduced to unity. "This idea," says Voltaire, "only shows that he delighted in every thing extraordinary and difficult." 10. Before taking leave of the subject of Arithmetical Scales, it seems proper to explain the method of transferring any numerical expression from one scale to another. When a number is expressed by any other scale, let it be required to express it on the Denary or common scale. Ac cording to the principle of notation, the value of any figure is increased as many times as there are units in the index of the scale, by each removal of place toward the left hand; hence the method is obvious. Ex. Transfer the number 2342 from the Quinary to the Denary scale. 2= 2 5×4 20 5x5x375 5x5x5x2=250 Ans. 347 on the Denary or common scale. Ex. 2. Express by figures on the Denary scale the number 7643 on the Octary. 3= 3 8×4 32 8x8x6 384 = 8x8x8x7=3584 Ans. 4003 on the Denary. Again, when a number is given on the Denary scale, let it be required to express it on any other scale. The method of doing this will be readily understood from an example. Ex. 1. Transfer the number 8576 from the Denary to the Senary scale. 6)8576 Ex. 2. Transfer the number 105463 from the Denary to the Quinary scale. Of the Grecian, Roman, and Modern Notation by 11. The Greeks expressed numbers by means of twentyseven distinct characters. Twenty-four of these were furnished by the letters of their alphabet, to which they added three additional ones to make up the number. The first nine letters denoted units, or counters on the first bar; the second nine, tens, or counters on the second bar; and the third nine, hundreds, or counters on the third bar. By writing a small dash below or above these characters, their value was increased a thousand or ten thousand times. 12. The principal characters which the Romans employed in their notation, and which are still in use for representing dates, and numbering chapters, &c. were I, V, X, L, C, D, M, by the various combinations of which, they were enabled to express any number. Probable origin of these characters. The most convenient, and even the most natural way of representing unity, by a written character, is by means of a stroke, 1; two, in like manner, by two strokes, II; three by three strokes, III; and so on, till the reckoner reached the number ten, when in order to intimate that the first bar, or that of units, was completed, he would throw a dash across the mark for unity, and thus might arise X, the mark for ten. Were this separated into two, from the point where the strokes intersect either of them, V would become the mark for the half of ten, or for five. When the reckoner arrived at the third bar, or that of hundreds, he would combine three strokes, E, which, from greater dispatch in writing, would, in process of time, assume a round form, C. This character, E, separated into two, would give For L, either of which L would represent fifty. In like manner, when he arrived at the fourth bar, or that of thousands, he would combine four strokes, M, which assuming a round form, would appear thus, M, or CD. This last, separated into two parts, would give CI or I, either of which I being more condensed into the form D, would signify the half of a thousand, or five hundred. The characters, CIO, and its half, ID, are sometimes used without any contraction; the former of which is increased ten times by writing on the right hand and C on the left; the latter ten times by writing on the right hand merely. Thus in regard to the increasing of C15, CID=1000 and CCI-10,000. Next in respect of the character 10, 10=500 and 100=5000. Farther, a stroke above any of these characters, increases its value a thousand times. Thus M=1000, and M=1,000,000. The rules by which the Romans expressed any number by those characters, whose origin we have endeavoured to trace, are the following: First, When the same character is repeated any number of times, its value is to be taken as often. Second, When a character of less value is placed on the left hand of one of greater value, the value is to be deducted from that of the greater. Lastly, When a character of less value is placed to the right of one of greater value, its value is to be added to that of the greater. Thus X=10, XX=20, and XXX=30. IV=4, IX=9, and XIX= 19. VI=6, XI=11, and LI=51. 13. The modern notation, which is far superior to that of either the Greeks or Romans, uses nine artificial characters expressive of number, as has been already alluded to, together with the cipher, which of itself implies a privation of va These characters are, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, which, with the exception of the cipher, are called digits, (a term derived from the Latin word digitus, signifying the finger,) in reference to the primitive and predominating custom of reckoning by the fingers. They were first introduced into Europe by the Moors, who are said to have borrowed them from the Indians. Hence, the modern or common notation is sometimes called the Indian. 14. In order to facilitate the writing or reading of numbers, the digits, by which they are expressed, are sometimes classed into periods of six figures each, beginning from the units place. To these periods distinct names are given, which are as follows: units, millions, billions, trillions, quadrillions, quintillions, sextillions, &c. These periods are farther divided into half periods of three figures each. The first half period is called the period of units, the second that of thousands throughout all the full periods. Thus the number 5,482,584,878,284,800 which expresses the number of square feet in the surface of our globe, is read 5 thousand 482 billions, 584 thousand 878 millions, 284 thousand 800 units. The French, in a manner somewhat simpler, divide numbers into periods of three figures each, which are called, respectively, units, thousands, millions, billions, trillions, &c. in creasing in a thousand fold proportion. Hence, the French and English modes of numeration agree as far as hundreds of millions, but the English billion is a thousand times greater than the French. 15. It is sometimes convenient to conceive the digits by which any number is expressed, as denoting units of various degrees of value. By the palpable symbols, for example, on the Denary scale, ten counters on the first bar are equivalent to one on the second, and ten on the second equivalent to one on the third &c. So in figurate arithmetic, ten units of the first order are equivalent to one of the second, and ten of the second equivalent, in like manner, to one of the third, &c. In this way, simple units are called units of the first order; tens, units of the second order; hundreds, units of the third order; and so on. 16. However numerous the several operations to which numbers are subjected may appear, still they are all reducible to those of conjoining and separating. When two or more numbers are conjoined, that is, formed or collected into one, which is called their sum, the process by which this is effected is denominated Addition. When one number is drawn, or separated from another, so as to leave their difference, the process by which this is accomplished is called Subtraction. If the numbers to be added are all equal to one another, the process is capable of being much abridged, and then assumes the name of Multiplication. If the Subtraction be limited to the continued withdrawing of the same quantity from another, it is then also capable of abbreviation, and takes the name of Division. ADDITION. 17. Addition, according to what is stated in the preceding article, is the method of finding a number which shall be equal to two or more given numbers. The addition of numbers may be effected by setting out from one of them, and counting forwards as many units as are contained in each of the others successively. Suppose it were required, for example, to find the sum of 7 units, 5 units, and 6 units. By adding unity five times to 7, we obtain in succession the numbers eight, nine, ten, eleven, and, lastly, twelve. Again, by adding unity six times to twelve, we arrive in like manner at eighteen, thus showing that the sum of the given numbers is 18. Were the numbers large, |