(1+r)t Hence, the aggregate present value of all the t payments is equal to Or, denoting the present value by p, and summing the geometrical series within the parenthesis, we have. t 1. To find the present value of an annuity of $100, to continue 21 years at 6 per cent. Under the heading, 6 per cent., and opposite the number 21, we find 11.76408, which, multiplied by 100, gives for a result $1176,41 for the required present value From what has been stated, it will be easy to find the present value of a deferred certain annuity. It is evident that the present value will be found by finding the present value, as though it were to commence immediately, and then (A); finding the present value up to the time at which it is to be entered upon, and taking the difference between them. Thus, if it were required to find the present value of an annuity deferred 5 years, and then to continue 5 years, at 5 per cent., the annual payment being $100, we find the present value of the annuity for 10 years, commencing at once, to be $772,17, and for 5 years, $432,94; hence, the present value of the deferred annuity is $339,23. which is an expression for the present value of an annuity of one dollar per annum for years at r per cent. This value of p may be calculated by giving to r suitable values, corresponding to the usual rates of interest, as .04, .045, .05, &c., and attributing to t every value from 1 up to any given number. Such a table is called an annuity table, and shows by inspection the present value of an annual annuity of one dollar, or one pound, or any other unit of money, for any number of years at the ordinary rates of interest. The following is such a table : Présent value of an Annuity of $1. value of a perpetual annuity is equal to the annual payment divided by the rate per cent. If a perpetual annuity is deferred, its present value may be found in the same manner as in the case of a deferred certain annuity. If we denote the number of years which the annuity is deferred by T, the present value till the end of that time is, from equation A, 6.20979 equal to 0.95238 0.94340 5.07569 4.91732 7.10782 6.80169 6.00205 7.01969 6.73274 7.78611 7.43.533 8.53020 8.11090 7.72173 7.36009 9.25262 9.95400 10.63496 8 76048 8.30641 7.88687 9.38507 9.98565 9.39357 8.85268 8.86325 8.38384 10.10590 11.27407 10.47726 18 13.75351 12.65930 11.68959 10.82760 19 14.32380 13.13394 12.08532 11.15812 12.82115 11.76408 II. To find the value of an annuity in ar 20 14.87747 13.59033 12.46221 11.46992 15.41502 14.02916 21 rears, or which has been forborne for t years. case in which payments are made m times There may be two cases: 1st. When the per year, we have only to recollect that the computation is made at simple interest: 2d. present value of such an annuity is the same At compound interest. as that of an annual annuity for mt years at 1. At simple interest. At the end of the first year, a payment a will be due, at the end of the second year, a second payment a will be due, together with ar, the interest on the first payment, and so on, as indicated below. At the end of 1st. year the sum due is a &c. 1+ m a + ar and equation (C) becomes =(1+1) m' T (1 + =) me. · (C'). 66 2. At compound interest. At the end of the first year, the payment a becomes due; at the end of the second year, the payment a becomes due, and the interest ar on the first payment; at the end of the third year, the payment a is due, and the interest r(2a+ar) upon the accumulated capital at the end of the second year, and so on as indicated below. Whole amount due at the end of 1st. year, a. 66 2d. 66 66 3d. 66 66 2a+ar=a+a(1+r). a {1+(1+r)+(1+r)2 or, summing the series and denoting the sum by S, III. LIFE ANNUITIES. When the annuity is to cease with the life of a certain individual or certain individuals, the computation becomes more complicated. It then becomes necessary to combine the results already obtained with the probabilities of the individuals, on the duration of whose lives it depends, surviving any given period. Now it has been shown, in discussing the theory of probabilities, that the measure of the probability of any event occurring, is the quotient obtained by dividing the number of favorable chances by the whole number of chances, both favorable and unfavorable. If, then, we denote the number of persons of a given age, who are living at a given period, by n, and the number of these persons who are living at the end of one year by k', the probability that any one of these will survive k! the year is ; if we denote the number who We have hitherto supposed the annuity k k''"' km! payable annually, but the principles which n n have been employed will be equally applicable that any one will survive three, four, . . . m to the case in which payments are made years. semi-annually, quarterly, or at any regular period of time. The values of k', k'', &c., are taken from extensive tables of mortality which have been To modify equation (A) so as to apply to a prepared to show the ratio of the number of There is no way of computing the value of whence this series except by finding from the data given the value of each term separately and then taking their sum. n (1+r) and if we make a = 1, P = k' n (1 + r) 1+ P' }· (D). However, as the object in general is not to determine the value of an annuity at any particular age, but to construct a table show- Hence, to find the present value of an annuity ing this value at every age, there is a method of 1 dollar, or other unit, which depends upon of deducing the value at one age in terms of the life of an individual aged A years, knowthe value at another age, which was dis- ing that of one depending the life of an indicovered by Euler, and which serves to abridge vidual aged A+ 1 years, add 1 to the last valthe operations when such a table is to be ue and multiply the sum by the probability of calculated. To explain this method, let us the life A lasting one year, into the present consider the case in which the life annuity value of a unit due at the end of 1 year. depends upon the life of an individual A years of age. If we denote the present value of such an annuity by P, we shall have, from what has just been shown, a k' k" + &c. + &c. } · (1). By the aid of this rule, extensive tables of life annuities have been calculated for every possible age. By the aid of these tables, a variety of problems in annuities may be solved. To find the present value of a deferred life annuity: suppose, for example, that the person on whose life the annuity depends, is 30 years of age, and that the annuity is deferred 7 years. After 7 years, if the person be thenue were it dependent only on the life of B. alive, the value of the annuity for the remain- Let p denote the probability that A will live der of his life, will be equal to that on the life of a person aged 37 years, which may be found from the tables. Let this be denoted by P. The present value of one dollar, certainly due at the end of 7 years, may easily be computed; designate this by P'"; denote the probability that a person aged 30 will live to be 37, by K, which number may be found from the tables of mortality, and it is evident that K.P"" will denote the present value of 1 dollar due 7 years hence, taking into account the chances of the life in question. If, therefore, we designate the present value of the deferred annuity by Q, we shall have Q = Kx P" x P". -- more than n years, and q the probability that This expression is the measure of the proba- a (1 + r)»' The present value of a temporary life annuity, to run n' years, may be found by adding together n' terms of the series in the second member of Equation (1); or, it may be found by taking the present value of the whole life value of k, gives annuity, and then subtracting from it the present value of the annuity in the same life deferred n' years, since the sum of the temporary life annuity and of the deferred annuity which constitutes the remainder of the life if a = 1, the expression becomes annuity, makes up the entire life annuity. 2. When the annuity depends upon the joint continuance of two lives. If we denote the probabilities that A will survive 1, 2, 3, &c. years, by k', k'', k'", &c., and that B will survive 1, 2, 3, &c. years, by h', h'', h'" &c., then from the theory of probabilities, the probability that both will survive 1, 2, 3, &c. years, will be k' h', k'' h'', k''' h'", &c., and from the principles already employed, we shall have for the present value of a joint life annuity, depending upon these lives in tion (1), and h' has an analagous value with respect to the life of the second individual, &c. a a p (p+qpq) = = (1 + r)" (1+r)" a q + (1+r)" a p q 9 + Pq + (1+r)2 + &c. P'q' p"q" +r (1+r)2 But the first term of the second member is the present value of an annuity on the single life of A, the second term of an annuity on the single life of B, and the third the value of an annuity on the joint lives of A and B. Hence, the present value of an annuity on 3. When the annuity depends upon the life of the surviving life of two, is equal to the sum of the survivor of two individuals. Let P denote the annuities on each of the single lives, diminthe present value of the annuity, were it de-ished by the annuity on their joint lives. pendent only on the life of A, and P' its val- Many other problems may arise in the dis cussion of annuities, but the principles indi- | AN-TI-LOG'A-RITHM. [L. anti and log cated above are sufficient to show the method arithm]. Is a number corresponding to any of solving them. When there are more lives than two, the number of different cases that may arise becomes very great. A complete discussion of them would require more space than can be given to this subject. ANʼNU-LAR. [L. annulus, a ring]. Something which has the form of, or which resembles a ring. Thus, if a square be revolved about a straight line parallel to one of its sides as an axis, it will generate an annular solid. AN'NU-LUS. A portion of a plane included between the circumferences of two concentric circles. AN'SWER. A solution; the result of a mathematical operation. The term is chiefly used in arithmetic and algebra. AN-TAG-O-NIST'IC. [Gr. avт, against, and aywviorns, a champion]. Acting against each other; as the antagonistic screws in the level and theodolite. [Gr. αντι, given logarithm. Thus, 100 is the antiloga- In this sense the AN-TI-PAR'ALLELS, in Geometry, are straight lines which make equal angles with two given straight lines, but in contrary order. F C E B ANT-ARC'TIC CIRCLE. against, and aрктоç, the bear]. A small cirThus, if AC and AB are two given straight cle of the celestial and terrestrial spheres, lines, and the two straight lines CB and ED which passes through the southern pole of are so situated as to make the angle DEA the ecliptic. It is distant from the equator equal to the angle ACB, and the angle EDA about 664 It takes its name from being equal to the angle B: then are the last two opposite to another circle, which passes lines antiparallels with respect to the first through the north pole of the ecliptic, called two, and conversely, the first two are antiparallels with respect to the last two. See Sub-contrary. the Arctic circle. AN-TE-CED'ENT. [L. ante and cedo, to go before]. Of a ratio, is the first of the two terms which are compared together. As its name implies, it forms the standard of comparison, since it must be known before the value of the consequent can be expressed. The measure of the ratio of the antecedent to the consequent is, therefore, the quotient which arises from dividing the latter by the former. AN-TI-CLI'NAL LINE. [Gr. avтi, against, and kλvw, to incline]. In topography, a line from which the surface dips in both directions at right angles to it. The crest of a hill or ridge is an anticlinal line. AN-TIM'E-TER. [Gr. avri and μέτρον, measure]. An optical instrument for measuring angles more accurately than can be done by means of the sextant. AN-TIP'O-DES. [Gr. avri, against, and rodos, foot]. Two points on the earth's surface at the extremities of the same diameter. They have the same latitude, the one north, and the other south, and are distant from each other 180° in longitude. A'PEX. [L. apex]. The vertex, top, or summit of any thing. The apex of a cone, or pyramid, is the same as its vertex. APO-THEM of a regular polygon, is the perpendicular distance from the centre to one of the sides of the polygon. If n denote the number of sides of the polygon, r the radius of the circumscribed circle, and a the apothem, we shall have |