The Young Man's Best Companion

so many tens, from 10 to 90. The number of hundreds were expressed by other letters, supplying what was wanting either by other marks or characters, or by repeating the letters with dif- . ferent signs in order to describe thousands, tens of thousands, &c. About the year of Christ 200, a new kind of arithmetic, called sexagesimal, was invented by Ptolemy. Every unit was supposed to be divided into 60 parts, and each of these into 60 others, &c. Thus from 1 to 59 were marked in the common way: then 60 was called a sexagesima, or first sexagesimal integer, and had one single dash over it, as I'; 00 times 60 was called 'sexagesima secunda,' and marked l", &c. These methods of calculation are continued by astrologers in the subdivisions of the degrees of circles. The decuple, or Arabian scale, substitutes decimal instead of sexagesimal progression, and by this single process removes the difficulties and embarassments of the preceding modes. Thus the signs of numbers, from 1 to 9, are considered as simple characters, denoting the simple numbers subjoined to the character; the cipher, 0, by filling the blanks, denotes the want of a number, or unit, in that place; and the addition of the columns in a ten-fold ration, always expressing ten times the former, leads from tens, according to the order in which they stand, in a method at once most luminous and certain.

For decimal parts we are indebted to Regiomontanus, who, about the year 1464, published his book of Triangular Ca nons.' Dr. Wallis invented the use of circulating decimals, and the arithmetic of infinites; but the last, and, with regard to extensive application, the greatest improvement which the art of computation ever received, was from the invention of logarithms, the honour of which is due to John Napier, baron of Merchiston, in Scotland, who published the discovery about the beginning of the seventeenth century. Mr. Henry Briggs followed Baron Napier on the same subject. Arithmetic may now be considered as having advanced to a degree of perfection, which, in former times, could scarcely have been conceived, and to be one of those few sciences which have left little room for further improvement. In NOTATION we must observe, that all numbers are expressed by these ten figures, following, viz.

] 2 3 4 5 6 7 8 9 0

Nine of these are called significant figures, to distinguish them from the cipher, which of itself signifies nothing; but, as it is placed in whole numbers, serves to increase the value of the next figure or figures that stand before it as 3 is but three; but before the cipher, thus 30, the 3 becomes thirty, &c. We are to note, That every one, or any of the above nine figures, or digits, have two values; one certain, and another uncertain: the certain value, as when it stands alone by itself; the uncertain is, when

joined or placed with other figures or ciphers; for when any one of these figures stands alone, it signifies no more than its own simple value; as 5 is but five, 4 but four, 6 but six, and 3 no more than three, &c. And this is the certain value of a figure. But when another figure or cipher is annexed, then they are increased in their value ten times; as 5, or five units or ones, to five tens, or fifty; 4 to four tens, or forty; and 3 to three tens, or thirty as thus, 51, fifty-one; 42, forty-two; 34, thirty-four, &c. Again, if any of the said figures stand in the third place towards the left hand, they signify so many hundreds as they expressed units or ones: as 500, is five hundreds; 400, four hundreds; and 300, three hundreds, &c. If any of them possess the fourth place towards the left hand, they are so many thousands as they contain units; and so any or every figure increases in a tenfold proportion, from the right hand to the left, according to the place it is found or stands in; so that five may be either five, or fifty; five hundred, or five thousand: in the first place, 5; in the second, 50; in the third, 500; in the fourth place, 5000, &c. The true value of figures in conjunction may be fully learnt and understood by the following table:

For the easier reading of any number, first get the words at

the head of the table by heart; as units, tens, hundreds, thousands, &c. and apply them thus: 75, five units, five, and 7 tens, seventy; that is seventy-five, Again 678; 8 units, eight; 7 tens, seventy; and 6 hundreds, six hundreds; that is, six hundred and seventy-eight. Once more, 3456; units, six; 5 tens, fifty; 4 hundreds, four hundreds; 3 thousands, three thousands; together, three thousand four hundred fifty six. The fourth line of the table, viz. 123456789, may be read thus: one hundred twenty-three millions, four hundred fifty-six thousand, seven hundred and eighty-nine. But the manner of reading any number may be rendered more intelligible by stops, thus: make a comma after every third figure or cipher, beginning at the right hand, and so on towards the left, thereby distinguishing every third place into hundreds; as hundreds of units, hundreds of thousands, hundreds of millions, and hundred-thousands of millions, &c. And, for trial, let us read the first line of the table: where the last place in valuation is hundred-thousands of millions, and being pointed into periods, will stand thus, 123,456,789,012, and is to be read thus; one hundred twenty-three thousand, four hnndred and fifty-six millions, seven hundred eighty-nine thousand, (no hundreds) and twelve. Again, the following number, viz. 276,245,678,921,460, is to be read thus: 276 million of millions, 245 thousand of millions, 678 millions, 921 thousands, 460 units or ones; that is, two hundred and seventy-six million of millions, two hundred forty-five thousand six hundred seventyeight millions, nine hundred twenty-one thousand, four hundred and sixty. The foregoing Table of Numeration is on the right hand distinguished into such periods, for the easier reading thereof. Numbers to be read or written, viz.

96, Ninety-six

242, Two hundred and forty-two

7924, Seven thousand 9 hundred 24 54006, Fifty-four thousand and six

524707, Five hundred 24 thousand 707

4706240, Four million 706 thousand 240

62700472, Sixty-two millions 700 thousand 472

474960204, Four hundred 74 millions 960 thousand 204 4214007042, Four thousand 214 millions 7 thousand 42 44214800240, Forty-four thousand 214 millions, 8 hundred thousand 240.

ADDITION.

Is the putting together two or more numbers, or sums, so as their total value may be discovered or known. Herein we must always observe to set the numbers to be added orderly one under

the other, that is, units under units, tens under tens, hundreds under hundreds, &c. as in the subsequent examples.

In addition of simple numbers, whether it be yards, gallons, pounds, or any thing else, remember to carry one for every 10 that you find in the right hand row, or rank of figures being units, to the next row of tens; and the like from the rank of tens to the row of hundreds, &c. and whatever it makes in the last row, you must set down, amount to what it will. Then in casting up each example, to know its total, I begin at the right hand, or unit's rank of the first example, and say, 2 and 4 is 6, and 6 is 12, and 8 is 20, aud 2 is 22, and 4 is 26: in which row there are two tens, and 6 over; wherefore I set down 6 just under its own rank, and carry two to the next row, and say, two that I carry and 4 make 6, and 2 is 8, and 8 is 16, and 6 is 22, and 4 is 26, and 2 is 28; and this being the last row, I set down the amount, viz 28 so that the total number of yards is found to be 286. And the amount of the next or second example, is found by the same method to be 3562 gallons. And in the third and last example, the total number of pounds is found by the same way to be 247484. And so the total of any other example of the same kind, viz. simple numbers of any denomination, may be found. Note, That when any of the ranks amount to just 10, 20, 30, 40, 50, &c. then you must set down the O under its proper rank, and carry either 1, 2, 3, 4, or 5, according to the number of tens that you may find, to the next row.

ADDITION OF MONEY.

4 farthings

12 pence
20 shillings