Article 17. Having the Dimensions of any of the parts of a Circle, to find the Side of a Square, equal to the Circle 20. Having the Side of a Square, to find the Diameter of a Circle, 21. Having the Side of a Square, to find the Circumference of a Cir- 22. Having the Diam. of a Circle, to find the Area of a Semicircle 456 23. Having the segment of a circle, to find the length of the Arch Line 456 24. Having the Chord and Versed Sine of a Segment, to find the Di- OF THE CHARACTERS MADE USE OF IN THIS TREATISE. THE sign of equality: as 12 pence = - 1 shilling, signifies that 12 pence are equal to one shilling; and, in general, that whatever precedes it is equal to what follows. + The sign of Addition: as 5+5=10, that is, 5 added to 5 is equal to 10. Read 5 plus 5, or 5 more 5 equal to 10. The sign of Subtraction: as, 12-4-8, that is, 12 lessened by 4 is equal to 8, or 4 from 12 and 8 remains. Read 12 minus 4, or 12 less 4 equal to 8. X The sign of Multiplication: as 6×5-30, that is, 6 multiplied by 5 is equal to 30. Read 6 into 5 equal to 30. or 5)30(The sign of Division: as 30÷5-6, that is, 30 divided by 5 is equal to 6. Read 30 by 5 equal to 6. 875 25 Numbers placed fractionwise, do likewise denote division, the numerator or upper number being the dividend, and the denominator or lower number, the divisor; thus, is the same as 875-25-35. 875 ::: The sign of proportion, thus, 2 : 4 :: 8: 16, that is, as 2 is to 4 so is 3 to 16. Signifies Geometrical Progression. 9-26-13 Shews that the difference between 2 and 9 added to 6 is equal to 13. Read 9 minus 2 plus 6 equal to 13. And that the line above (called a Vinculum) connects all the numbers over which it is drawn. 12-344-5 Signifies that the sum of 3 and 4 taken from 12 leaves or is equal to 5. 2 Signifies the second power, or Square. 3 Signifies the third power, or Cube. m Signifies any power in general, as 62=square of 6; and 503-cube of 50, &c. thus m signifies either the square or cube, or any other power, ✔, or Prefixed to any number or quantity, signifies that the square root of that number is required. It likewise (as also the character for any other root) stands for the expression of the root of that number or quantity to which it is prefixed. As 366, and 108436—12, and 361-6, &c. 3 ors Prefixed to any number, signifies that the cube root of that number is required, or expressed. As 216-6, and 5134-2169, &c. or 216] 216* = 6, &c. 3 3 Signifies any root in general. As 3613 square root, 216] 3eube n Thus, signifies either the square root, cube root, or any oth root whatever. abcd When several letters are set together, they are supposed to be multiplied into each other; as those in the margin are the same as axbXcxd, and represent the continual product of quantities or numbers. If a be the root, then a Xaaa or a2 is the square of a, and a×a×a=aaa or a3 is the cube of c, &c. Note. The figure above is called the index of the power. It is usual to write shillings at the left hand of a stroke, and pence at the right; thus, 13/4 is thirteen shillings and four pence. Note. The use of these characters must be perfectly understood by the pu pil, as he may have occasion for them. A NEW AND COMPLETE SYSTEM OF ARITHMETICK. UNIV. OF RITHMETICK is the Art or Science of computing by num Air, ETC is both in Theory and Practice. The Theo ry considers the nature and quality of numbers, and demonstrates the reason of practical operations. The Practice is that, which shews the method of working by numbers, so as to be most useful and expeditious for business, and is comprised under five principal or fundamental Rules, viz. NOTATION OF NUMERATION, ADDITION, SUBTRACTION, MULTIPLICATION, and DIVISION; the knowledge of which is so necessary, that scarcely any thing in life, and nothing in trade can be done without it. NUMERATION 1. TEACHES the different value of figures by their different places, and to read or write any sum or number by these ten characters, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.0) is called a cypher, and all the rest are called figures or digits.* The names and significations of these characters, and the origin or generation of the numbers they stand for, are as follow; O nothing; 1 one, or a single thing called an unit; 1+1-2, two ; 2+1=3, three; 3+1=4, four; 4+1=5, tive; 5+1-6, six; 6+17, seven; 7+18, eight; 8+1=9, uine; 9-+1=10 ten; which has no single character; and thus, by the continual addition of one, all numbers are generated. 2. The value of figures when alone, is called their simple value, and is invariable. Besides the simple value, they have a local value, that is, a value which varies according to the place they stand *These figures or digits were obtained from the Arabians, and were introduced into Europe in the ninth century. The Arabs probably derived the decial notation from India. The fexagefimal divifion had previously been in general ufe in Europe. This mode of divifion is yet retained in a few cafes, as in the divifion of time, where fixty minutes make an hour, fixty feconds a minute, &c. The figures are doubtless called digits from digitus, a finger, because counting ufed to be performed on the fingers. C in when connected together. In a combination of figures, reckoning from the right to the left, the figure in the first place represents its simple value; that in the second place, ten times its simple value, and so on; each succeeding figure being ten times the value of it in the place immediately preceding. There is no reason in the nature of numbers that their local value should vary according to this law. They might have been made to increase in 3, 4, 5, &c. fold, or in any other ratio. The tenfold increase is assumed because it is most convenient. 3. The values of the places are estimated according to their or der: The first is denominated the place of units; the second, tens; the third, hundreds; and so on, as in the table. Thus in the num ber-5293467; 7, in the first place signifies only seven; 6, in the second place, signifies 6 tens, ur sixty; 4, in the third place, four hundred; 3, in the fourth place, three thousand; 9, in the fifth place, ninety thousand; 2, in the sixth place, two hundred thousand; 5, in the seventh place, is five millions; and the whole, taken together, is read thus; five millions, two hundred and ninety three thousand, four hundred and sixty seven. The process of Numeration may be more clearly seen by the following Six places of figures, beginning on the right, are called a period, and each successive six places another period. Each period is considered as divided into two half periods of three figures each. These are distinguished by the comma, and the point for a period. There is an obvious reason for this division into periods, for at the beginning of each period, there is a new denomination of units, of which the tens, hundreds, thousands, &c. are numerated as in the first period. 4. A cypher, though it is of no signification itself, yet, it possesses a place, and, when set on the right hand of figures, in whole .Thousands numbers, increases their value in the same tenfold proportion; thus, 9 signifies only nine; but if a cypher is placed on its right hand, thus, 90, it then becomes ninety; and, if two cyphers be placed on its right, thus, 900, it is nine hundred, &c. 5. To enumerate any parcel of figures, observe the following Rule. First, commit the words at the head of the table, viz. units, tens, hundreds, &c. to memory, then, to the simple value of each figure, join the name of its place, beginning at the left hand, and reading towards the right. More particularly-1. Place a dot under the right hand figure of the 2d, 4th, 6th, 8th, &c. half periods, and the figure over such dot will, universally, have the name of thousands. -2. Place the figures, 1, 2, 3, 4, &c. as indices over the 24, 3d, 4th, &c. period. These indices will then shew the number of times the millions are increased. The figure under 1, bearing the name of millions, that under 2, the name of billions (or millions of millions) that under 3, trillions. EXAMPLE. Sextillions. Quintilli. Quatrill. Trillions. Billions. Millions. Units. th. un. th. un. th. un. th. un. th, un. th. un, c.x.t.c.x.U. 6 5 4 3 2 1 913,208,000,341,620,057,219,356,809,379,120,406,129,763 NOTE 1. Billions is substituted for millions of millions: Trillions, for millions of millions of millions; Quatrillions, for millions of millions of millions of millions, and so on. These names of periods of figures, derived from the Latin numerals, may be continued without end. They are as follows, for twenty periods, viz. Units, Millions, Billions, Trillions, Quatrillions, Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Fredecillions, Quatuordecili ions, Quindecillions, Sexdecillions, Septendecillions, Octodecillion, Novemdecillions, Vigintillions. Fifteen. THE APPLICATION. Write down, in proper figures, the following numbers. Two hundred and feventy nine. Three thoufand four hundred and three. Thirty feven thousand, five hundred and fixty seven. Nine millions, feventy two thousand and two hundred. 15 279 37567 9072200 2543431702 |