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Now far in the noon-heated air
And toward a sombre palace-tomb
A dusk procession treads in gloom;
Round the cattle on the bank-
The brassy gad-fly, keen and lank,
But toward the funeral train afar
“ Methinks, a simple grave of reeds,
Is nearer to the morning star
Beside the stream, throngh every change
Unto her small and snowy tent,
The sandy path spread eastward far,
Where dwelt, beside a low bright star,
And as across the desert dim
6 Methinks,” she said, “my heart can tell
And raise as sweet a hymn-
CURIOSITIES OF SCIENCE.
THE DENARY SCALE.
Has it ever occurred to the reader to enquire why it is that when we count as far as ten we return again on the same digits, and count ten-and-one, or eleven ; ten-and-two, or twelve ; ten-and-three, or thirteen ; and so on to twice ten, or twenty; ten-times ten, or a hundred ; ten-times a hundred,
l or a thousand, &c.; every conceivable number being, in fact, only some combination of ten and of the nine digits which precede it ? The matter is well worth enquiry, although, like many other common things," it is
too farniliar to excite our curiosity; and it becomes still more curious when we consider that this mode of counting is universal. Without any possibility of deriving it from each other, the people of all times and all nations have adopted it. Navigators tell us, indeed, of some few isolated savages who, when first discovered, were unable to reckon higher than five; but this fact can scarcely be considered as making any material exception to this general rule. It may be asserted, then, that all mankind have, by a common consent or common instinct, agreed to count by ten, or, in other words, have adopted the denary or decimal scale of numbers—so called from the Latin words, deni and decem, ten—and hence has this scale been regarded as an institution of nature. Men have received from nature ten fingers and ten toes, and there can be no doubt that these appendages of the human body have suggested the base or radix, as it is called, of the depary system. In fact, the nine first numbers with the cipher (0) are called digits, from the Latin word digitus, which signifies either a finger or a toe.
But some people will have it that although nature taught us, as it were, to count by tens, still that the system is by no means the best that could be devised. Thus, they say, if we had a binary. scale, in lieu of the existing one-that is, if we counted by twos, and if every number were only a combination of one and two, or rather of 1, and the cipher which with 1 would then represent two, in the same way as 1 and 0 now do ten, all our arithmetical calculations would be wonderfully facilitated, except so far as the lengthy shape which numbers would assume ; for our present number 1,000, for instance, would then be represented as 1111101000.
The celebrated Leibnitz, and some others, recommended the binary scale as the most convenient; but mathematicians generally are agreed that if the adoption of a radix or base of notation were an arbitrary arrangement that could be ma over again, the best possible one would be 12; that is, the substitution of a duodenary (or duodecimal) scale for the common or denary one ; so that we should count in twelves instead of tens, and any number on being removed one place more to the left hand, instead of being increased ten times in value, as at present, should be increased twelve times.
The adoption of the duodenary scale would require two additional characters, to represent 10 and 11 as simple digits; and it is usual to employ two letters of the Greek alphabet for that parpose, as # and Q. Thus the digits of this scale would stand thus, viz. :--1, 2, 3, 4, 5, 6, 7, 8, 9, T, Q, 10; the highest of the digits, or the radix, in any arithmetical scale being represented by a unit prefixed to a cipher, as we now represent ten.
The chief advantage which the duodenary scale would possess over the common scale would be found in fractional numbers, or in those fractional forms equivalent to our decimals, a great many of which that are now indefinite would then be definite. This would arise from the fact that while 10, the radix of the common scale, has only two factors, i. e., 2 and 5, 12, the radix of the duodenary scale, bas four, or is divisible by four numbers, viz., 2, 3, 4, and 6; and the fraction whose denominator would be any of these factors, or any multiple of any of them, would be definite in the form equivalent to our decimals. Thus, the decimal equivalent to fin
the common scale is •3333 &c., to infinity, without ever arriving at the truth, whereas the same fraction expressed on the same principle in the duodenary scale is simply ·4.
The French, by the use of the metrical system, adapt all their comp!itations admirably to the common or denary scale; whereas for many of our subdivisions of coins, weights, and measures, the duodenary scale would be more convenient. Thus our shilling is divided into 12 pence; our pound troy into 12 ounces; our foot into twelve inches, and these again into 12 lines. Hence, it is not without some purpose that a chapter is devoted in many of our books of arithmetic to the duodenary system of notation, or duodecimals, as they are commonly termed; and it frequently happens that where feet and inches are to be multiplied by feet and inches, as in finding the superficial or cubic content of bodies, the operator finds it more convenient to reduce bis numbers to duodecimals, and having performed the multiplication, to reduce the result back again into the common notation.
Many of the curious properties which belong to the number 9, in the common system, would be transferred to the number 11, in the duodecimal scale. Thus, as multiplication and division are proved by the operation called “casting out the 9's,” in the common aritbmetic, they would be proved by casting out the 11's in duodecimal arithmetic.
The arrangment by which any number when moved one place to the left hand is increased tenfold in value, when moved two places is increased a hundred fold, and so on by successive powers of ten—this arrangement, we say, is another thing in our common system of notation, which is apt to appear to us so simple as not to be worth while stopping to enquire about. Some people, no doubt, if they think about it at all, imagine that it must have been always so, and cannot conceive any system of arithmetic without it. But, nevertheless, it was not so always, and the mathematicians of ancient Greece and Rome were obliged to conduct all their calculations without having the slightest idea of this admirable system of notation. Who the inventor of this notation was is a question beyond the reach of investigation. We know that it was introduced into Europe along with our present numerical characters, by the Arabs, in the tenth century, and that it was obtained by them from the Indians, whence it is known as the Indian system of notation ; but beyond this all enquiry fails, altbough the invention, whoever its author may have been, is admitted to be one of the most useful and important improvements ever introduced into mathematical science,
Pope Sylvester II., a Frenchman,* who died in 1003, and who learned
* This extraordinary man is also well known as Gerbert, Archbishop of Rheims. He studied under a Spanish bishop in Catalonia, and while in Spain acquired his knowledge of Arabian mathematical learning. It is a fact interesting to Irishmen that he was for a great part of his life abbot of the famous Irish abbey of Bobbio, founded in northern Italy by our great St. Columbanus. Muratori attributes the first revival of learning in Italy to Gerbert's school at Bobbio ; and there is no doubt that the same great man introduced science
to himself from the Moors in Cordova, was the first who introduced this system if arithmetic into France; and it was brought into England many years later by Athelard, a monk of Bath, who flourished in 1127, and was the first to introduce into England “Euclid's Elements," which be translated into Latin out of Arabic. According to Wallis, Athelard, indeed, shares the honour of having introduced the Arabic, (or rather Indian,) arithmetic into England, with two or three other Englishmen, who travelled into Spain about the same time, as Robert of Reading, who translated the Koran into Latin, in 1143, and Daniel Morley, who studied mathematics and the Arabic language at Toledo, about 1180.
To appreciate the advantages which we derive from this Indian system of notation, it is only necessary to consider for a moment the system which preceded it.
With only ten numerical symbols we are able to express any conceivable number, whereas the Greeks were under the necessity of using 36 different characters, and even with these were, for many centuries, unable to express any number greater than 10,000, or a myriad. The characters employed by the Greeks were principally those of their alphabet. Thus the nine digits were represented by nine letters of the alphabet; dine other letters were used to express 10, 20, 30, &c.; and nine others for 100, 200, 300, &c. For the thousands, 1000, 2000, 3000, &c., instead of distinct characters, they had recourse again to those for the simple units, with the addition of an iota or dash below the letters, and so on up to 10,000, higher than which they could not go, until Archimedes thought of adding the letter M to any other number to multiply it by 10,000; and further of putting the squares, cubes, or other powers of a lesser number to indicate a number of very great magnitude. His method was still further improved by Appollonius, who, about two centuries before Christ, so perfected the Greek notation, as to have apparently reached within a single step of the Indian notation, which, however, he was not fortunate enough to discover.
The Roman numerals are familiar to every one, as they are still used very frequently for dates, inscriptions, &c. The capital letters I, V, X, L, C, D, and M, signify respectively the numbers 1, 5, 10, 50, 100, 500, and 1,000; but if the reader substitutes the Roman for the common numerals in any lengthy questions of multiplication or division, he will find the operation both tedious and troublesome; while, of course, the higher arithmetic of decimals and logarithms would be altogether impossible.
It is a remarkable circumstance that the denary, or decimal system of numbers, although not the result of man's learning or forethought, is nevertheless, according to the admission of mathematicians, at least the next best that could be adopted. We say this because when ten was judged by the Creator as the suitable number of fingers for the hand of man, it does not Germany. He was the first Frenchman who ascended the papal chair. He was, undoubtedly, the most learned man in Europe in his time; and his enemies attributed his learning, as well as his elevation to the pontificacy, to the practice of magic.
follow that it should be equally suitable for the calculations of science, which was to be man's own creation—the work of his own proper intelligence yet even for this purpose it is admitted to be also admirably adapted.
And it is also a curious circumstance, that we are indebted to some nameless sage, perhaps of Central Asia, for a system of notation infinitely more perfect than any which the philosophers of ancient Greece and Rome were able to devise--for one, in fact, to which is due all the perfection to which modern analytical science has arrived at.
MASTERS OF MANY LANGUAGES. We feel naturally a greater attraction to a man who understands many languages than we do to great chemists, botanists, physiologists, metaphysicians, or the like. The latter may demand our respect for great attainments, or great services, still we are drawn by a certain wonder, love, and longing to the eminent linguist. For he speaks to us and to many men, uttering languages to us unknown-he thus forms a living link between beings of the same form, but dissimilar speech—he, in fact, for a moment, and very partially, effaces the results of the presumption of Babel, and makes us think of the happy time, of which we are certified not less by science than by Scripture, when all the world was of one language and one speech. And a memory of happiness is sure to awaken feelings of kindness, if not gratitude, to him who is the occasion of it,
Chief amongst the celebrated names of antiquity we find that of Mithridates, king of Pontus, and enemy of Rome, His name is familiar to every schoolboy who has dipped a little in the Latin writers. His empire comprised, according to some writers, twenty-two-according to others, twentyfive different nations; it is asserted that he was perfectly acquainted with the languages spoken by these peoples, Ancient writers assert that he never needed to employ an interpreter in conversation with his subjects, though speaking tongues so dissimilar. The famous Cleopatra is likewise stated to have spoken with ambassadors from several lands, in their own languages. It is not asserted, however, that she at all equalled the great king of Pontus in a knowledge of languages, as certainly it could not be expected, than any lady could at all peer his tremendous capacity for eating and drinking!
In the middle ages there occasionally shone men renowned for their knowledge of languages; the dispersion of the Greeks, the intrusion of the Moors into Spain, the Crusades-all such movements, which cast intellectual men into contact with new peoples, naturally led rapidly to the increase of their knowledge of languages. Roderigo Ximenes, Archbishop of Toledo, in the thirteenth century, is stated to have preached to congregations Romans, Germans, French, English, Navarrese, and Spanish, addre