Because the number of figures in the given square number is even, we find the nearest square number to the two first figures 29, which is 25, the root whereof 5, we set in the quotient, and deduct 25 from 29, and to the residue 4, we annex the following figures, 81, so we have 481, for a resolvend. The double of the first figure in the quotient being 10, is then set as a divisor to 48, all the figures in the resolvend, but the last; and finding it to be contained 4 times, we annex the 4 to the divisor and quotient; the then divisor, 104, is multiplied by the last figure in the quotient, 4, and the product 416 is deducted from the resolvend 481, to the residue whereof is annexed the two following figures in the square, so we have 6516 for a new resolvend, to all which figures but the last we make 108, the double of 54, the figures in the quotient a divisor, and finding it will be contained 6 times, we place 6 in the divisor and quotient; the then divisor 1086 is multiplied by the last figure in the quotient 6, and the product being set under the resolvend and thence deducted, leaves nothing; so is 546 the root sought. For if the root 546 be squared or multiplied by 546, the product will be the square number given. 546 quotient 546 74 44 Here being an odd decimal figure, we annex any odd number of cyphers to make the decimal places even; and then extracting the root as before, we thence cut off half the number of decimals that wo have in the square. Thus, If to the square of this root we add the remaining figures 20551, we shall have our given square, whose root was required. What is the square root of 16007.3104? What is the square root of 348.17320836 ? What is the square root of 12345678987654321 ? The application of this will hereafter be shewn, THE ELEMENTS OF PLANE GEOMETRY. DEFINITIONS. GEOMETRY is that science wherein we consider the properties of magnitude. 2. A point is position without magnitude, as A. fig. 1. 3. A line is length without A -B breadth, as AB. fig. 1. and 2. 4. The extremities of a line are points, as the extremities of the line AB are the points A and B. 5. A right line is the shortest that can be drawn between any two points, as the line AB. If it be not the shortest, it is then called a curve line, as AB. fig. 2. 6. A superfices, or surface, is that which has length and breadth, without thickness, as ABCD. fig. 3. 7. The extremities of a superfices are lines. 8. The inclination of two lines meeting one another, or the opening between them, is called an angle. Thus fig. 4. the inclination of the line AB to the line BC meeting each other in the point B, or the opening of the two lines BA and BC, is called an angle, as ABC. Note, when an angle is expressed by three letters, the middle one is that at the angular point. 10. When the lines that form the angle are right ones, it is then called a right-lined angle, as ABC. fig. 4. If one of them be right and the other curved, it is called a mixed-angle, as B. fig. 5. If both of them be curved it is called a curved line or a spherical angle, as C. fig. 6. 11. If a right line, CD. fig. 7. stand upon another right line, AB, so as to make the angles ADC, CDB, on each side, equal to each other, these angles are called right angles, and the line CD a perpendicular. 12. An obtuse angle is that which is greater than a right one, as the angle ADE, fig. 7. and an acute E |