its brevity and precision; instead of employing twenty four characters, only nine digits and a cypher are wanted. The symbols also are more simple, more appropriate and determined; and therefore the powers of them are less liable to inaccuracy or confusion. With the symbols too, the scale of numerical calculations has been varied. The first improvement was the introduction of reckoning by tens, which, no doubt, took its rise from the obvious mode of counting by the fingers, as that was customary in the primary calculations of every nation, except the Chinese.

The Greeks, as will be seen in the following lessons, had two methods of marking the advance of numbers: one on the plan which was afterwards adopted by the Romans, and which is still used to distinguish the chapters and sections of books; and in the other, the first nine letters of the alphabet represented the first numbers from 1 to 9, the next nine so many tens, from 10 to 90. The number of hundreds was expressed by other letters, supplying what was wanting either by other marks or characters, or by repeating the letters with different signs in order to describe thousands, tens of thousands, &c.

Upon this mode of computation, the writers in a modern encyclopedia have the following judicious and apposite observations. "The ancient Greeks and Romans would have brought the science of arithmetic to a much greater degree of perfection than they ever did, had they hit upon the method of expressing by ten distinct characters the numbers by which they reckoned. But the idea of a cypher, which can only be introduced into the decadary system, and which may be styled the KEY STONE of Arithmetic, seems never to have struck them; and thus, though they reckoned properly enough by tens, yet not having characters proportionate enough to express their numbers, they involved their Arithmetic in a labyrinth of confusion, from which neither a EUCLID, nor an ARCHIMEDES, with all their wonderful mechanical powers, were able to extricate it, for want of this clue. In a word it is to the cypher, in uniform alternation with the nine digits, that the moderns owe the honour of having perfected a science, in which the ancients, with all their great attainments, had made but small progress. And perhaps if all our modern weights and measures were divided and subdivided upon the decadary plan, instead of into fourths, eighths, twelfths, sixteenths, &c. that general uniformity of both, so long wanted, might soon be attained."

About the year of our Lord 200, a new kind of arithmetic, called sexagesimal, was invented by Ptolemy. Every unit was

L

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 l'; 60 times 60 was called " sexagesima secunda," and marked 1", &c.

These methods of calculation are continued by astronomers 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 embarrassments 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 cypher O, 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 ratio, 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 Canons. Dr. Wallis invented the use of circulating decimals, and the arithmetic of infinities; 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 his 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 little room for improvement.

LESSON THE FIRST.

1. Arithmetic is the science of numbers, and teaches the art of computing by them.

2. The Greeks made use of the letters of their alphabet to represent their numbers. The Romans followed

the same method; and besides characters for each rank of classes, they introduced others for five, fifty, and five-hundred, &c.

3. Any number may be represented by repeating and combining the letters: thus xx stands for two tens, or twenty: ccc for three hundred, and so on.

4. When a numeral letter is placed after, one of greater value, their values are added: thus XII stands for ten and two, or twelve: LXXVII for seventy-seven: MDCLXVI, for one thousand six hundred and sixty-six.

5. When a numeral letter is placed before one of greater value, the value of the less is taken from that of the greater: thus Ix stands for ten less one, or nine: XL for fifty less ten, or forty: xc one hundred less ten, or ninety.

6. The method of notation that we now use is taken from the Arabians, and the characters by which all the operations of common arithmetic are performed, are, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0; the first nine are called significant figures.

7. Significant figures, when placed singly, denote the simple numbers subjoined to the characters.

8. When several significant numbers are placed together, the first or right hand figure only is to be taken for its simple value: the second signifies so many tens, the third so many hundreds, and so on.

Ex. In 7777, the right hand figure stands for 7 only, the next stands for 70, the third for 700, and the fourth for 7000, so that the whole reads seven thousand, seven hundred, and seventy-seven.

9. The cipher in any place denotes the want of a number in that place, thus 50 denotes five tens, and no unit or simple number: so 304 denotes three hundred and four, there being no significant figure in the tens place.

10. The whole art of arithmetic is comprehended in various modifications of the four rules, Addition; Subtraction; Multiplication; and Division.

11. Addition is that operation by which several nnmbers or sums are collected into one total.

12. Subtraction is the operation by which we take a less number, or sum, from a greater, and find their difference.

13. Multiplication is a compendious mode of addition, and teacheth to find the amount of any given number, by repeating it any proposed number of times. £. S. d.

14. Division teacheth to find how often one number is contained in another of the same denomination, and thereby performs the work of many subtractions.

Example. 8)76543

£. S. d. 6) 865 14

9

QUESTIONS FOR EXAMINATION.

1. What is arithmetic?

2. What did the Greeks and Romans use f numbers? 3. How were numbers represented by the Roman letters? 4. In what way was the addition of a number performed? 5. How was a number taken away by the Roman method? 6. From whom is derived the modern method of notation, and what are the characters used?

7. What do the significant figures denote when placed singly? 8. How are they reckoned when placed together?

Explain by the example.

9. What does the cipher denote?

10. In what is the whole art of arithmetic comprehended? 11. What is addition?

Explain by the examples.

12. What is subtraction? Explain by the examples?

13. What is multiplication, and what does it teach?

Explain by the examples.

14. What is division, and what does it teach?

Explain by the examples.

LESSON THE SECOND.

1. Reduction teaches to bring numbers from one denomination to another, without changing their value: it is used to simplify the operations in other rules.

Example.-If I wish to know how many half-crowns there are in £1. 12s. 6d. I reduce the given sum, and also the half-crown into pence, or into sixpences, and divide the greater by the less: thus there are 5 sixpences in half a crown, and 65 sixpences in £1. 12s. 6d. and 65 divided by 5 give 13 for the answer, that is, there are 13 half-crowns in £1. 12s. 6d.

2. In reduction all great names are brought into lesser by multiplication: and less names are brought into greater by division.

Example.-Pounds are brought into pence by multiplying by 20 and by 12: and pence are brought into guineas by dividing by 12 and 21.