Cento

“ Several nations in America have no other instruments of calculation. It was probably the same in the primitive ages. It is another strong presumption of the truth of what I now advance, that all civilized nations count by tens, tens of tens, or hundreds, tens of hundreds, thousands, and so on; still from ten to ten. We can discover no reason why the number ten should be chosen rather than any other for the term of numeration, except this primitive practice of counting by the fingers. ”

“ While considering the extent of the savage's arithmetical knowledge, of his ability to count and to grasp the meaning of number, it may not be amiss to ask ourselves the question, what is the extent of the development of our own number sense? To what limit can we absorb the idea of number, with a complete appreciation of the idea of the number of units involved in any written or spoken quantity? Our perfect system of numeration enables us to express without difficulty any desired number, no matter how great or how small it be. ”

“ It is thought there may be five thousand gentlemen; every gentleman, taking one with another, may afford to swear an oath every day, which will yearly produce one million, eight hundred, twenty-five thousand oaths, which number of shillings makes the yearly sum of ninety-one thousand, two hundred and fifty pounds.
The farmers of this kingdom, who are computed to be ten thousand, are able to spend yearly five hundred thousand oaths, which gives twenty-five thousand pounds; and it is conjectured, that from the bulk of the people twenty, or five-and-twenty thousand pounds may be yearly collected. ”

“ Ten, according to some, is a perfect number, not in its composition like six, but because it contains inclosed in itself all the simple numbers and all the properties, as of even and odd, perfect and imperfect, and it is the beginning of all numbers composed of tens, and also for the foundation of our law. ”

“ Besides, if a hundred is [Pg 358] sometimes used for totality, as when the Lord said by way of promise to him that left all and followed Him, "He shall receive in this world an hundredfold;" [704] of which the apostle gives, as it were, an explanation when he says, "As having nothing, yet possessing all things," [705] —for even of old it had been said, The whole world is the wealth of a believer,—with how much greater reason is a thousand put for totality since it is the cube, while the other is only the square? ”

“ As an instance the word cent exists in every language, but cent in French consists only of four letters placed side by side one after the other, and would never be anything else to us if we could not count; but to count is to add and to take away, that is to say, addition and subtraction, thus to conceive and name; in order to possess a hundred objects, it does not suffice to see them only, it is necessary to count them up to the hundred. ”

“ A fresh mental effort is required of the ignorant child of nature; and the result is “all the fingers and one more,” “both hands and one more,” “one on another count,” or some equivalent circumlocution. If he has an independent word for 10, the result will be simply ten-one. When this step has been taken, the base is established. The savage has, with entire unconsciousness, made all his subsequent progress dependent on the number 10, or, in other words, he has established 10 as the base of his number system. ”

“ Giles's Church at Edinburgh obeyed an impulse to which millions of the human race may be permitted to remain strangers. But the prescriptions of reason are absolute, unchanging, of universal validity; to count by tens is the easiest way of counting—that is a proposition of which every one, from here to the Antipodes, feels the force; at least I should say so if we did not live in a country where it is not impossible that any morning we may find a letter in the Times declaring that a decimal coinage is an absurdity. ”

“ You now see that the convenience of our method of numeration arises from a few simple signs being made to change their value as they change the column in which they are placed. The same advantage arises from counting in a similar way all the articles which are used in every-day life. For example, we count money by dividing it into pounds, shillings, and pence, of which a shilling is 12 pence, and a pound 20 shillings, or 240 pence. We write a number of pounds, shillings, and pence in three columns, generally placing points between the columns. ”

“ This limit is often extended to 20, and not infrequently to 100. Again, we find 1000 as the limit; or perhaps 10,000; and sometimes the savage carries his number system on into the hundreds of thousands or millions. Indeed, the high limit to which some savage races carry their numeration is far more worthy of remark than the entire absence of the number sense exhibited by others of apparently equal intelligence. If the life of any tribe is such as to induce trade and barter with their neighbours, a considerable quickness in reckoning will be developed among them. ”

“ Forming, then, a series of fractions whose denominators are the products of the number of persons indicated in the table of mortality as living at the age of that one who settles the annuity, by the successive powers of unity augmented by the rate of interest, and whose numerators are the products of the annuity by the number of persons living at the same age augmented successively by one year, by two years, etc., the sum of these fractions will be the amount required for the life annuity at that age. ”

“ In transforming a vulgar fraction, say 1/17, to a decimal [Pg 229] fraction, we proceed as below, adding as many noughts to the dividend as we like until there is no remainder, or until we get a recurring series of figures, or until we have carried it as far as we require, since every additional figure in a never-ending decimal carries us nearer and nearer to exactitude.
Now, since all powers of 10 can only contain factors of the powers of 2 and 5, it clearly follows that your decimal never will come to an end if any other factor than these occurs in the denominator of your vulgar fraction. ”

“ A student in a library of a thousand volumes, an officer before his regiment of a thousand men upon parade, may have a pretty clear idea of the units as well as of the totals; but when we come to a thousand times a thousand, or a thousand times a million, all view of the units, for most men, probably for every man, is lost: the million for the grown man is in a great degree like the hundred for the child. ”

“ The proportion consecrated may be a certain ratio of income fixed on a sliding scale, on the principle that the greater the profits, the greater the proportion which me be spared. For instance, on the first day of each week, or month, or quarter, or year, one may consecrate a certain proportion of his profits of that week, month, quarter, or year to the Lord, say five, eight, or ten per cent., in case they rise to a specified amount ”

“ When one number is increased by others, the number which is as large as all the numbers together is called their sum. The process of finding the sum of two or more numbers is called Addition, and, as was said before, is denoted by placing a cross (+) between the numbers which are to be added together.
Suppose it required to find the sum of 1834 and 2799. In order to add these numbers, take them to pieces, dividing each into its units, tens, hundreds, and thousands:
1834 is 1 thous. 8 hund. 3 tens and 4;
2799 is 2 thous. 7 hund. 9 tens and 9.
Each number is thus broken up into four parts. ”

“ When the savage begins the process of counting he invents, one after another, names with which to designate the successive steps of his numerical journey. At first there is no attempt at definiteness in the description he gives of any considerable number. If he cannot show what he means by the use of his fingers, or perhaps by the fingers of a single hand, he unhesitatingly passes it by, calling it many, heap, innumerable, as many as the leaves on the trees, or something else equally expressive and equally indefinite. ”

“ Few tests of relative stages of civilisation are more trustworthy than that of the definite conception of high numbers. The prevalence of a decimal system of numerals among widely severed nations, alike in ancient and modern times, has been universally ascribed to the simple process of counting with the aid of the fingers. ”

“ If you once commit the addition table thoroughly, you have it fixed for life; whereas if you do not, you have to make the calculation over again every time, and thus lose a vast amount of labor. I have no doubt that there are some in this class who are in the habit of counting, who have ascertained that seven and eight for instance, make fifteen; by counting up from seven to fifteen, hundreds of times. ”

“ One million is a large number! "If you had a million shillings to count one by one, and did it as fast as you could for ten hours a day, it would take a fortnight; and the million shillings would weigh nearly five tons, or be a heavy load for a railway truck." ”

“ If the consultant has a lucky number, and this appears with good signs, it promises much success. An unlucky number with gloomy signs predicts misfortune. A journey with a five near obviously points out that it will be taken in five days, or weeks, and so on. Ten dots, close together, means ten pounds or shillings, according to the size and number of the dots. Numbers with the symbol of a legacy show the amount to be expected.
Nun.—This is a sign that you will probably remain unmarried through your own choice; to the married it implies unjust suspicion. ”

“ On the divergence of the basis itself and the confusion caused—for it is five, or it is six, or ten or twelve, or sixty or a hundred, or three hundred and sixty or the double hundred, the thousand. The reason, not clearly understood by the people, is that Six and the Dozen are first, and Sixty is a compromise in condescending to the people. ”

“ The first Appendix, on Computation, and the sixth, on Decimal Money, should be read and practised by every student with as much attention as any part of the work. The mastery of the rules for instantaneous conversion of the usual fractions of a pound sterling into decimal fractions, gives the possessor the greater part of the advantage which he would derive from the introduction of a decimal coinage. ”

“ Sometimes a number above ten, but not over twenty, is indicated with the hands only by counting up to ten in the ordinary way, and then opening all the fingers and counting again, until they reach the requisite amount in excess of ten. ”

“ The sum of two thousand pounds a-year is reckoned a noble fortune in France, and very, very few, there are that possess that sum.
One thousand pounds a-year constitutes a handsome fortune for a gentleman; and four hundred for a bourgeois, or for one employed in trade or commerce. ”

“ How, then, is work perfected in the number forty? The reason, it may be, is, because the law was given in ten precepts, and was to be preached throughout the whole world: which whole world, we are to mark, is made up of four quarters, east and west, south and north, whence the number ten, multiplied by four, comes to forty. ”

“ A curious Recreation with a Hundred Numbers, usually called the Magical Century.
If the number 11 be multiplied by any one of the nine digits, the two figures of the product will always be alike, as appears from the following example:—
Now, if another person and yourself have fifty counters apiece, and agree never to stake more than ten at a time, you may tell him, that if he will permit you to stake first, you will always undertake to make the even century before him. ”

“ For instance, that sum gives thirty-two letters per head to the population in the year; and, of course, as thousands of us can’t write, and thousands more don’t write, it follows that the real correspondents of the kingdom do some pretty stiff work in the writing way. But these are only the letters. If you include somewhere about four hundred and twenty million post-cards, newspapers, book-packets, and circulars, you have a sum total of fourteen hundred and seventy-seven million odd passing through our hands. ”

“ If there is this tendency to adopt 100 as a basis instead of 10, the teaching of decimals might sometimes be simplified by proceeding from percentages to percentages of percentages, i.e. by commencing with centesimals instead of with decimals.
22. Perception of Number.—In using material objects as a basis for developing the number-concept, it must be remembered that it is only when there are a few objects that their number can be perceived without either counting or the performance of some arithmetical process such as addition. ”

“ Mademoiselle Marguerite seemed inclined to volunteer some remark, but the doctor imposed silence upon her by a gesture, and continued his examination. “Is the count a great eater?” he inquired. “Does he drink heavily?”
“The count is moderation itself, monsieur, and he always takes a great deal of water with his wine.” ”

“ It is, however, correct to say, "The signers of the petition were less than a hundred," where the round number, a hundred, is something like a collective noun, and less is thought of as meaning a less quantity or amount. ”

“ If the strong man who had six fingers on each hand, and six toes on each foot, had invented a system of numeration, it would have gone in twelves, nearly like the duodecimals which our carpenters use; unless, indeed, he had been stupid after the manner of very strong men, and not gone beyond sixes. We see how the Romans, though they inherited from their Eastern ancestors a numeration by tens up to decem, and then beginning again undecim, &c., yet when they began to write a notation could get no farther than five—I., II., III., IV., V. ”