3)2343 8) 781 2 23606, which is the square root of 5 true to the last figure. And thus any power or root may be extracted. The ultimate convergency depending on the numbers v and a, which is here 1, and not upon the indices; the series will, however, converge slower, the higher the root is to be extracted. 97 4)485 8)121 15 7 5)105 8) 21 2 Problem 3. A series being given to find the value of a, the first factor, so that it may obtaip a given convergency at the xth term. Find the convergency of the series, and for % substitute its value in terms of x, and the first term a; then make the formula, thus altered, equal to the given convergency, and the value of a will be found by this equation. General Erample. 1 Given the general series +... to find the ya cebat cato die of a, so that the series may obtain a given convergency 1 at the s term of the series. Now the general convergency is and the value of a in a+2(m-1) the x term is x(1-1)+a; which, being substituted in the general % -% a+ (0-2) convergency gives i then the resolu. 2+ (m-1) a+?(m+x2) tion of the equation atx(x-2) 7 gives az a+%(m+1-2) s(m+3-2)-zl(2-2) bal Particular Example. To find the first factor a in the series 1 1 1 so that it may obtain a convergency of f in the fourth term. x(m+x2)-5b(x-2) Here z=2, m=5, x=4, b=3; therefore 2(5+4-2)-2.3(4—2), =1; and therefore the series 1 3-) 1.3.5.7.9 1 + + &c. will obtain a convergency of } in the fourth term. 3.3.7.9.11 General Erample. + ادا 2% The series +B -&c. being given, to (@+ 2x)R le-4 (a+z)R find the value of a, so as to make the convergency in the sth term. 2 then putting %(31) The convergency of this series is expressed by [2(t-1)+a]R z(c-R)-1) R Problem 4. Two series which do not converge at the same rate being given, to find the number of the term, if possible, in which their convergencies Find the convergency of each series ; then make the two expreso sions equal ; and the value of x being found, that of ¢ will be obo tained from the equation z=3(x-1ta. are equal. 1 1.30x it 13.30-&c., and +B Or, find the convergency of each series in terms of x and a, the first factor, and make the two expressions equal; then the value of : in the equation will give the number of the term. Example. 1 The two series + 5.32' 9.3* 1 1.4 2.4 34 A -C + &c. being given, -viAR (a+4)R (a +8)R (a +12)R to find x the number of the term when their convergencies are equal. 41—8ta The convergency of the first of these series is 9[4(3-1)+a] 4(1-1) and that of the second is i therefore, putting these R[4(1-1)+a] 41-8ta expressions equal to one another, thus, 4(x-1) R[ 4(-1)+a)941a-1)+a] 4(1-1)_41-8+a aR+36-BR gives x = for the number of R 9 36-4R the term, as required. Let a=33 ; then will x=59. or THE ELEMENTS OF EUCLID. BOOK I. Definitions. 1. A POINT is that which hath no parts, or which hath no magnitude. 2. A line is length without breadth. 4. A straight line is that which lies evenly between its extreme points. 5. A superficies is that which hath only length and breadth. 6. The extremities of a superficies are lines. 7. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. 8. A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction." 9. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. N. B.' When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of those straight lines, and the other upon the other line : Thus the angle which is contained by the straight lines AB, CB, is named the angle ABC, or CBA; that which is contained by AB, BD, is named the angle ABD, or DBA; and that which is contained by DB, CB, is called the angle DBC, o. CBD; but if there be only one angle at a point, it may be expressed by a letter placed at that point ; as the angle at E.' 10. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it. 11. An obtuse angle is that which is greater than a right angle. 12. An acute angle is that which is less than a right angle. 13.“ A term or boundary is the extremity of any thing." 1 14. A figure is that which is inclosed by one or more boundaries. 15. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another. 16. And this point is called the centre of the circle. 17. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. 18. A semicircle is the figure contained by a diameter and the part of a circumference cut off by the diameter. 19. “ A segment of a circle is the figure contained by a straight line, and the circumference it cuts off.” 20. Rectilineal figures are those which are contained by straight lines. 21. Trilateral figures, or triangles, by three straight lines. 22. Quadrilateral, by four straight lines. 23. Multilateral figures, or polygons, by more than four straight lines. 24. Of three-sided figures, an equilateral triangle is that which has three equal sides. 25. An isosceles triangle is that which has only two sides equal. 26. A scalene triangle, is that which has three unequal sides. 27. A right-angled triangle, is that which has a right angle. 28. An obstuse-angled triangle, is that which has an obstuse angle |