interest of 11. for 1 year, gives the value of the perpetuity. So, if the rate of interest be 5 per cent. = Then 100a ÷ 5 = 20a is the value of the perpetuity at 5 per cent: Also 100a 4 25a is the value of the perpetuity at 4 per cent: And 100a 3: 33a is the value of the perpetuity at 3 per cent: and so on. = Again, because the amount of 1. in n years, is R", its increase in that time will be R" -1; but its interest for one single year, or the annuity answering to that increase, is R1; therefore as R — 1 is to R" 1, so is a to m; that RAI is, m X a. Hence, the several cases relating to Annuities in Arrear, will be resolved by the following equations: RA In this last theorem, r denotes the present value of an annuity in reversion, after 1 years, or no commencing till after the first years, being found by taking. the difference between the two values - X years and years. But the amount and present value of any annuity for any number of years, up to 21, will be most readil£ he LE TABLE I The Amount of an Annuity of 17. at Compound Interest. Yrs. at 3 perc. 34 perc. 4 per c. 14 per c. 5 per c. 6 per c. 14 15 16 17 18 19 20 21 15.6178 16-1130 16-6268 17:1599 17-7130 18-8821 17 0863 17 6770 18-2919 18.9321 19:5986 21.0151 20 3236 20-7841 21-5786 23-2760 21-8245 22-7193 23-6575 25 6725 21-7616 22-7050 23 6975 24 7417 25-8404 28 2129 23-4141 24 4997|| 25-6454| 26-8551| 28-1324 309057 25-1169 26 3572 27-6712 29 0636 30 5390 33-7600 26.8704 28-2797 29-7781 31-3714 33 0660| 36 7856 28.6765 30 2695 31.9692 337831 35 7193 39-9927 TABLE II. The present Value of an Annuity of 12. Yrs at 3 erc 134 per c. 4 per c. 143 per c. | 5 per c. 6 per c. To find the Amount of any annuity forborn a certain number of years. TAKE out the amount of 17. from the first table, for the proposed rate and time; then multiply it by the given annuity; and the product will be the amount, for the same number of years, and rate of interest.-And the converse to find the rate or time. Exam. To find how much an annuity of 507. will amount to in 20 years, at 3 per cent. compound interest. On the line of 20 years, and in the column of 34 per cent. stands 28-2797, which is the amount of an annuity of 17. för the 20 years. Then 28.2797 X 50 gives 14139851. = 1413. 198. 8d. for the answer required. To find the present Value of any annuity for any number of years. Proceed here by the 2d table, in the same manner as above for the 1st table, and the present worth required will be found. Exam. 1. To find the present value of an annuity of 501, which is to continue 20 years, at 3 per cent.-By the table, the present value of 17. for the given rate and time, is 14 2124; therefore 14 2124 × 50 = 710-621. or 7101. 128. 4d. is the present value required. Exam. 2. To find the present value of an annuity of 207. to commence 10 years hence, and then to continue for 11 years longer, or to terminate 21 years hence, at 4 per cent. interest. In such cases as this, we have to find the difference between the present values of two equal annuities, for the two given times; which therefore will be done by subtracting the tabular value of the one period from that of the other, and then multiplying by the given annuity. Thus, tabular value for 21 years 14·0292 ditto for - 10 years 8.1109 GEOMETRY. 1. DEFINITIONS. A POINT is that which has position, but no magnitude, nor dimensions; neither length, breadth, nor thickness. 2. A Line is length, without breadth or thickness. 3. A Surface or Superficies, is an extension or a figure, of two dimensions, length and breadth; but without thickness. 4. A Body or Solid, is a figure of three dimensions, namely, length, breadth, and depth, or thickness. 5. Lines are either Right, or Curved, or Mixed of these two. 6. A Right Line, or Straight Line, lies all in the same direction, between its extremities; and is the shortest distance between two points. When a line is mentioned simply, it means a Right Line. 7 A Curve continually changes it direction between its extreme points. 8. Lines are either Parallel, Oblique, Perpendicular, or Tangential. 9. Parallel Lines are always at the same perpendicular distance; and they never meet though ever so far produced. 10. Oblique lines change their distance, and would meet, if produced on the side of the least distance. 11 One line is Perpendicular to another, when it inclines not more on the one side than than the other, or when the angles on both sides of it are equal. 12. A line or circle is Tangential, or a Tangent to a circle, or other curve, when it touches it, without cutting, when both are produced. 13. An Angle is the inclination or opening of two lines, having different directions, and meeting in a point. 14 Angles are Right or Oblique, Acute or Obtuse. 15. A Right Angle is that which is made by one line perpendicular to another. Or when the angles on each side are equal to one another, they are right angles. 16. An Oblique Angle is that which is made by two oblique lines; and is either less or greater than a right angle. 17. An Acute Angle is less than a right angle. 18. An Obtuse Angle is greater than a right angle. 19. Superficies are either Plane or Curved. का L 20. A Plane Superficies, or a Plane, is that with which a right line may, every way coincide. Or, if the line touch the plane in two points, it will touch it in every point. But, if not, it is curved. 21. Plane figures are bounded either by right lines or curves. 22. Plane figures that are bounded by right lines have names according to the number of their sides, or of their angles; for they have as many sides as angles; the least number being three. 23. A figure of three sides and angles is called a Triangle. And it receives particular denominations from the relations of its sides and angles. 24 An Equilateral Triangle is that whose three sides are all equal. 25. An Isosceles Triangle is that which has two sides equal. 26. A |