EXAMPLES. 1. What is the Square Root of this Square Number 298116. 29,81,16(546 25 104)481 1086).6516 Because the Number of Figures in the given Square Number is even, we find the neareft Square Number to the two firft Figures 29, which is 25, the Root whereof 5, we fet in the Quotient, and deduct 25 from 29, and to the Refidue 4, we annex the following Figures 81, fo we have 481 for a Refolvend. The double of the firft Figure in the Quotient being 10 is then fet as a Divifor to 48, all the Figures in the Refolvend but the laft; and finding it to be contained 4 Times, we annex the 4 to the Divifor and Quotient; the then Divifor 104 is multiplied by the laft Figure in the Quotient 4, and the Product 416 is deducted from the Refolvend 481, to the Refidue whereof is annexed the two following Figures in the Square, fo we have 6516 for a new Refolvend, to all which Figures but the laft we make 108, the double of 54, the Figures in the Quotient a Divifor, and finding it will be contained 6 Times, we place 6 in the Divifor and Quotient; the then Divifor 1086 is multiplied by the laft Figure in the Quotient 6, and the Product being fet under the Refolvend and thence thence deducted leaves Nothing: So is 546 the Root fought.. For if the Root 546 be fquared or multiplied by 546, the Product will be the fquare Number given. 2. What is the Square Root of 1710864 ? Root. 1,71,08,64(1308 Answer. I 23).71 69 2608).20864 What is the Square Root of 3857-3? 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 we have in the Square. Thus, Root. 3857.300000(62.107 Answer. 36 122)257 244 1241)1330 124207) 890000 .20551 If If to the Square of this Root we add the remaining Figures 20551, we shall have our given Square, whofe 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 THE ELEMENTS MENTS O F PLANE GEOMETRY. Plate I. G DEFINITION S. EOMETRY is that Science wherein we confider the Properties of Magnitude. 2. A Point is that which has no Parts, being of itself indivisible, as A. 3. A Line has Length but no Breadth, as AB. Figures 1 and 2. 4. The Extremities of a Line are Points, as the Extremities of the Line AB are the Points A and B. Figures 1 and 2. 5. A right Line is the shortest that can be drawn between any two Points, as the Line AB. Fig. 1. but if it be not the shorteft, it is then called a curve Line, as AB. Fig. 2. 6. A Plate I. 6. A Superficies or Surface is confidered only as having Length and Breadth, without Thickness, as ABCD. Fig. 3. 7. The Extremities of a Superficies are Lines. 8. The Inclination of two Lines meeting one another (provided they do not make one continued Line) 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 expreffed 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 Mix'd-Angle, as B. Fig. 5. If both of them be curved it is called a Curvedlined or a Spherical Angle, as C. Fig. 6. 11. If a right Line CD (Fig. 7) fall upon another right Line AB, fo as to incline to neither Side, but make the Angles ADC, CDB on each Side equal to each other, then those Angles are called right Angles, and the Line CD a Perpendicular. 12. An obtufe Angle is that which is wider or greater than a right one, as the Angle ADE, Fig. 7. and an acute Angle is lefs than a right one, as ÉDB. Fig. 7. 13. Acute |