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ANNO ANNO DOMINI, the Year of our Lord, or the æra ANNOTO, a river of Jamaica, forming a bag of the ANNO DOMINI. of time from the birth of Jesus Christ. This is usually same name. It takes a northerly course, and enters the TO.

inserted in all public acts and writings of this country, sea between the rivers Blowing and Palmito. ANNO

ANNOUNCE', v. with the addition of the year of the king's reign.

Annuncio : ad, nuncio; to NOUNCE. TATE.

ANNONA CIVILIS, in Antiquity, corn, or provi ANNUNCIATE, >bring something new.
sions of any sort for a year. The tax called by this AnxUNCIATION. To make known; to publish,
name, intended to supply the capital and army, was to declare, to proclaim.
very oppressive in the time of Justinian.

Lo Sampson, which that was annunciat
ANNONA Militaris, was a name for necessaries

By the angel, long or his nativitee:

And was to God Almighty consecrat, stored in the magazines, for the maintenance of an And stode in noblesse while he mighte see. army during a campaign. We read also of annonæ

Chaucer. The Monkes Tale, vol. č.p. 139. præfectus, or curators, to inspect the sale of corn, an

Of thy birth at length, nonæ structor, to attend to the provision for the army, Announc'd by Gabriel, with the first I knew, annonarius, an oflicer who had the distribution of the And of the angelic song in Bethlehem field, rations to the soldiers, and annonarii, monopolists.

On thy birth-night that sung the Saviour born.

Milton's Par. Reg. Book ir.
Annona, in Botany, a genus of plants, belonging to

I will not cavil with antiquity, or traduce the primitive church, the class Polyandria, and order Polyginia.

but I think I may believe without danger, that those sibyls might ANNONAY, a town of Algiers, 32 miles from Con be select instruments to announce the dispensations of heaven to stantina. It is now only remarkable for some ancient


Howell's Letters.
ruins, on an elevated spot not far distant from the

Surely, if the plain man would ply his almanack well, that alone
ANNONDY, a town of France, in Lower Languedoc, viour. If one day teach him another, all days would teach him,

would teach him gospel enongh, to show him the history of his Sa121 leagues from Privas. It is now the head of a There should he see his Blessed Saviour's conception anminciated canton, in the arrondissement of Tournon, and was for by the angel: March 25. merly the capital of the Upper Vivarois, giving the

Bp. Halls Serinons. title of a marquisate to the house of Rohan-Soubise. When the revolntion of the anniversary calls on us to perform At present it is chiefly remarkable for its manufacture glorious benefits of Christ's Incarnation, Nativity, Passion, Resur

our duty of special meditation, and thankfulness to God for the
of paper of a very excellent quality, which was origin- rection, and Ascension, then we have the offices of Christmas, the
ally condncted by the celebrated aeronaut Montgolfier, Annunciation, Easter, and Ascension.
who ascended from this place in his balloon, on the Taylor's dpo. for Autho. 8. set Forms of Liturgie. Pref.
5th of June, 1783.

Those, mighty Jove, mean time, thy glorious care,
ANNOT, a town of France, in the department of the

Who model nations, publish laws, announce
Lower Alps, eight leagues from Digne. It is the head

Or life or death, and found or change the empire.

Prior. Hymn of Callimachus. of a canton, and contains a population of upwards of

Her [Queen Elizabeth's] arrival was announced through the coon: 1,000 inhabitants.

try by a peal of cannon from the ramparts; and a display of fireAN’NOTATE, v. Annoto, from ad, noto, which

works at night. ANNOTATION, Vossius thinks is from the su

Gilpin's Tour to the Lakes of Cumberland, &c. ANNOTA'TIONIST, pine Notum: for we note or ANNUNCIATION DAY, in Ecclesiastical Affairs, a feast

A.V'XOTATOR. mark a thing, that from the of the church, celebrated annually on the 25th of mark we may know it.

March, in honour of the salutation of the Blessed To make marks, or remarks or observations.

Virgin, or as some authors hold, of our Saviour himAt length hee [M. Tyndall] beethought hym selfe of Cutbert self. Bingham assigns the institution of this festival Tunstall then Bysliop of London, and especially for the great com

to the seventh century, about which time the council mendatio of Erasmus, who in his annotations so extolieth him for his learning, The Whole Workes of Wm. Tyndall, &c.

of Toledo ordered it to be celebrated eight days before (HENRY SAVILE] carefully collected the best copies of books, Christmas. Several Romish writers bring forward a written by St. Chrysostome, from various parts of the world, and spurious sermon of St. Athanasius, and another of Gre. employed learned men to transcribe, and make annotations on, gory Thaumaturgus, to prove its still greater aptithem.

If it [philology] be only criticism upon ancient authors and lan- quity. The eastern and western churches vary consi-
guages, he must be a conjuror that can make those moderns, with derably in their season of observing this feast. The
their comments, and glossaries, and annotations, more learned than Syrian calendar notes it down for the 1st day of De-
the authors themselves in their own languages, as well as the sub-cember, and distinguishes it by the appellation of
jects they treat.
T'emple, on An. and Mo. Learning.

Bascarach, inquiry, or investigation. The Greeks,
There are some admirable remarks in the annotations to the se-
cond chapter [of Dr. Lister's treatise De Condimentis & Opsoniis celebrate it even in Lent; while the Armenian

who are by no means scrupulous in its solemnization, Veterum], concerning the dialogue of Asellius Sabinus, who introduces a combat between mushrooms, chats, or beccoticos, oysters, churches, in order to prevent it from occurring at that and redwings; a work that ought to be published: for the same period, hold it on the 5th of January. The pope, at annolator observes, that this island is not destitute of redwings, one time, was in the habit of having a certain number though coming to us only in the hardest weather, and therefore of young maidens presented to him on Annunciation seldom brought fat to our tables.

Of his (Theobald's) notes I have generally retained those which Day, clothed in white serge from head to foot. To he retained himself in his second edition, except when they were

those who chose to be married by him, he gave 50 confused by subsequent annotators, or were too minute to merit crowns as a portion, those who chose to be devoted as preservation.

Johnson's Pref. 1o Shakespeare. nuns, received 100 crowns. The term Annunciation
Give me leare to annotate on the words. Ilive's Oralions.
How titly the Saracens are resembled to locusts, or scorpion- of the Jewish passover, in which the reason and origin

is also applied to designate that part of the ceremony
tailed locusts, in the Apocal. Mr. Mede has with far more clear-
ness sliewn, than the annotationists of the new way.

of its celebration are explained, called by the Jews Worthington's Miscellanies. 0721 Haggada, or the Annunciation..,


ANNOY', v.
It. Noia, Span. Enojo, Fr.Ennuy. ANNUAL, n.

ANNOY', n. From the Lat. Noxia (Noxia, nos AN'NUAL, adj.
ANNOY'ANCE, >cia, noia, Menage), from Noceo, AN'NUALLY,

Annuus, from Annus, a year.
ANNOY'FUL, to hurt.


Yearly, occurring every year. ]
To hurt, harm, or injure, to ANNU'ITANT,
trouble or molest.

And þe Romaynes beß anyed of here trauail so sore,

Of perel on se, & eke on lond, þat heo nul come her no more. For he hathe every zeer of annuelle Rente 300,000 Hors charged

R. Gloucester, p. 100. with Corn of dyverse Greynes and of Ryzs: and so helede the a fulle
Salomon sayth, that right as mouthes in the shepes fleese anoien noble Lif, and a delycate, aftre the custom of the Contree.
to the clothes, and the smale wormes to the tree, riglit so anoieth

Sir John Maundeville, p. 376. sorwe to the herte of man.

He ordeyned ye annual vse or ceremonie to eate the Paschall
Chaucer. The Tale of Melibæus, vol. ii. p. 74. Lambe, with whose bloude they sprynkeled the thrasholde and
But the cheer of the lord is on men that doen yuelis, and who is haunse of the dore, with both the postes of the house : and trusting
it that schal anoye you if ye ben sureis and loueris of goodnesse.

vpon this signe, feared not themselfes in the middles of the slaugh-
Wiclif. Peter 1. c. iii. ter of the Egipcians.
But telleth me your grefe,

Udall. Paul to the Hebrues, cap. xi.
Paraventure I may in your mischefe

There must be Masses dyrges, ther must be anuaries bead mē.
Conseile or helpe: and therfore telleth me

Bales' Image to both Churches, p. 91.
All your annoy, for it shal ben secree.

Wherfore first the officers seruauntes, wer put out of the Courte,
Chaucer. The Shipmannes Tale, vol. ii. p. 35. and many old officers were put to liue in their countreys, but the
For he was nevere crucyfyed, as thei seyn ; but that God made kyng (Henry the eighth] of his bountie enhaunsed their liuynges,
him to stye up to him with outen Dethe, and with outen Anoye. for he that had three pound wages, hal sixe pound annuitie, without

Sir John Maundeville, p. 162. attendaunce, and he that had_xl.s had foure pound, and so euery The lions which against other are of fiercenesse inuicible, they man after that rate, and young menne were put in their romes, either vanquished, or proued harmles, as though their mouthies

Hall. Henry VIII, fo. 146.
being stopped, or els their clawes fast bownden, they had had no In London was a preest, an annuellere,
power to hurt those whom god would haue preserued withoute any That therin dwelled hadde many a yere,
Udall. Paul to the Hebrues, c. xi.

Which was so plesant and so servisable
For al be it so, that al tarying be anoiful, algates it is not to re-

Unto the wif, ther as he was at table, preve in yeving of jugement, ne in vengeance taking, whan it is That she wold suffer him no thing to pay suffisant and resonable.

For borde ne clothing, went he never so gay.
Chaucer. The Tale of Melibaus, vol. ii. p. 77.

Chaucer. The Chanones Yemannes Tale, vol. ii. p. 244.
But certes ye han sodeinly cleped to your conseil a gret multi Get all the town to help, that will be hir'd,
tude of peple, ful chargeant and ful anoyous for to here.

Their pains I'll turn to annual holiday,
Id. ib. p. 94.

If it shall chance, but one bring word of her.
The city of Epidamnus became great and populous; and having

Beau 8. Fletch. Love's Pilgrimage, act v.
for many years together been annoyed with sedition, was, hy a war, Intelligent of seasons, and set forth
as is reported, made upon them by the confining Barbarians, Their aery caravan, high over seas
brought low, and deprived of the greatest part of their power.

Flying, and orer lands, with mutual wing,
Hobbes' Thucydides.

Easing their flight; so steers the prudent crane
And how he slew with glauncing dart amisse

Her annual voyage, borne on winds.
A gentle hynd, the which the lovely boy

Milton's Par, Lost. Book vii.
Did love as life, above all wordly blisse:

My grandfather had seven sonns, of which my father was the
For griefe whereof the lad n'ould after joy ;

youngest: to the eldest he gave his whole estate, and to the rest,
But pynd away in anguish and selfewild annoy.

according to the custome of those times, slight annuities. Spenser's Faerie Queene. Book i. c. vi.

Memoirs of Col. Hutchinson.
K. H. My Lords, at once. The care you have of us

Egypt, though there seldom falls any rain there, yet bath abun-
To mow down thorns that would annoy our fort.

dant recompense made it by the annual overflowing of the river.
Is worthy praise.

Ray. On the Creation. Shakespeare's 2d pt. H. VI. act iii. sc. 1. The outer and inner bark of trees serve to defend the trunk and Bart. No. Know the gallant monarch is in arms;

boughs from the excesses of heat and cold, and drought, and to And like an eagle o'er his aiery lowers

convey the sap for the annual augmentation of the tree.

To souse annoyance that comes near his nest.
Id. K. John, act v. sc. 2.

Ere the progressive course of restless age

Performs three thousand times its annual stage, Indeed though Stiff-CLAY (commonly called Stukley) be the

May not our power and learning be supprest, name but of one or two villages in the midst, yet their nature is extensive all over the country, consisting of a deep clay, giving much

And arts and empire learn to travel west?

Prior's Solomon. Book i. annoyance to passengers.

In short, oaths are the children of fashion; they are in some sense
Fuller's Worthies. Huntingdonshire.

almost annunls, like what I observed before of cant-words; and I
Say, what can more our tortured souls annoy,

myself can remember about forty different sets. Than to behold, admire, and lose our joy?

Swift's Introduction to Polite Conversation.

Prior's Pastoral. Trees receive annually their peculiar liveries, and hear their The very exercise of industry immediately in itself is delightful, proper fruits.

Wollaston's Religion of Nature. and hath an innate satisfaction, which tempereth all annoyances,

Supply anew and even ingratiateth the pains going with it. Barrow's Sermons.

With annuary cloaks the wandering Jew. Preserving his secret unrevealed, and his forces well united, let

John Hall's Poems. a hero march and annoy his enemy; for hot iron may form an

If the consent of the annuitants be requisite for every taxation, union

with hot iron; so he by equal fierceness, at a time when his they will never be persuaded to contribute sutriciently even to the foe is fierce, may conclude a firm peace.

support of government; as the diminution of their revenue must in Sir Wm. Jones's Hitópadésa.

that case be very sensible, would not be disguised under the apANNUA PENSIONE, in Ecclesiastical Affairs, an

pearance of a branch of excise, or customs, and would not be

shared by any other order of the state, who are already supposed ancient writ for providing the king's unpreferred chap- to be taxed to the utmost.

Thume's Essays. lains with a pension. Where an annual pension was due An annuity is a thing very distinct from a rent-charge, with to the king, from an abbot or prior, by this writ he could which it is frequently confounded : a rent-charge being a burthen nominate any of his chaplains (who were not provided yearly suni chargeable only upon the person of the grantor.

imposed upon and issuing out of land3, whereas an annuity is a with livings) to receive the same of such parties.

Blackstone's Commentarics,


AN It is our intention, under this general head, to treat means of our recent laudable institutions of saving. All. NUITIES. of all those subjects which have an obvious and neces banks, have all the advantages of compound interest, NUITE

sary dependence on the same principles of investiga- upon the same terms as the rich fund-holder who aptions, viz. annuities, certain and contingent, survivor- propriates a part of each of his dividends in the purships and assurances; we shall thus have the advantage chase of new stock. We might, therefore, without of saving numerous references to tables, formulæ, and much impropriety, pass over entirely the consideration theorems; unavoidable when these articles are treated of annuities at simple interest ; but as this article of in the places assigned to them in the alphabet by might thus appear incomplete, we shall briefly allude their initial letters.

to the subject, and then proceed to examine other Different 1. The doctrine of annuities has always been con cases of more practical utility. kinds of

sidered a subject of considerable importance in all wellannuities. regulated states; but in no country is it of so much con

Amount of annuities at simple interest.
sequence as in our own; and at no time did it possess 3. It may be proper to observe here, once for all, ANTE
such interest, even in England, as at the present ; when that we shall direct all our investigations, unless the same
property, either real or nominal, has attained to a contrary be specified, to those cases where the annuity
magnitude far beyond what could ever have been con is ll. and we may then, by a simple proportion, or
templated by the most sanguine financier, and the multiplication, determine the same for any other an-
transfer of it from one hand to another is the business nuity, as the amounts will evidently have the same
of every day, and the concern of almost every person ratio as the annuities themselves.
in the higher and middling classes of society.

This being premised, let
The term annuity is to be understood here to denote s = the amount of the annuity,
any periodical income arising from money lent, or from a = the annuity,
houses, lands, salaries, pensions, &c. payable from

n = the number of

years, time to time, either annually, or at any other interval. r = the rate of interest per l. per annum, These may be divided into such as are certain, and Now, in the case of the annuity of ll. per annum, such as depend upon some contingency, as the con it is evident that the amount for i year is 1 +r; for tinuance of one or more lives : these latter are called

2 years, 1 + 2 r, for 3 years, 1 + 31, &c.; and for life annuities. Annuities certain may likewise be di

n years, 1 t nr. And therefore the total amount for vided into such as are in possession, and such as are in n years, will be expressed by the series reversion ; the former signifying those that have already i+ (1+r) + (1+2r) + (1+3r),+, &c. 1 + (n-1), commenced, and the latter, those that will not commence till after some particular event, or till some

because it is to be observed, that for the last payment given period of time has elapsed.

no interest will be obtained ; and that when the an. With respect to the contingencies on which an an

nuity is for n years, the first sum received will only be nuity may depend, they are to be computed separately, at interest for (n − 1) year. upon the principles of the doctrine of probabilities,

This series is obviously an arithmetical progression, which shows the value of any given expectations whose first term is 1, the common difference 1, and founded upon the tables of mortality which have been

the number of terms n; moreover, the last term is 1 +
kept at different times and in different places. This (n-1) r; the sum of it will therefore be found by
subject is consequently attended with greater dificulty the usual rule, to be
than the former, although they both depend ultimately

(9 + n-1.r)n n (n-1)

upon the same fundamental principles; we shall, there-


2 fore, first solicit the reader's attention to the doctrine from which we deduce the following theorems, whence of annuities certain ; and afterwards pass to those which are contingent.

any one of the quantities may be determined when the

others are given.
§ I. Annuities certain.

=a}n +
n (n-1).r} = the amount ;

2. The principal questions relating to the doctrine

of these kinds of annuities, may be divided into two
parts ; viz. those relating to the amount, and those re-

= the annuity;

2n + n (n − 1)"
lating to the present value; and these will again evi-
dently differ according as simple or compound interest V{(2—-)2+8r%;}–(2+)
is used in the calculation. The value of annuities,

= No of years;
however, at simple interest, can hardly be considered
in the present day in any other light than as a mere
matter of speculation; so many ways presenting them-
selves by which compound interest may be obtained ;
even the savings of the simple artizan, amounting, per-

= rate of interest. haps, to not more than a shilling a week, may, by n (n − 1)


S = a

2 s

a =



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[ocr errors]


(1 + r)"–1;




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(1 +7) "=1).

; at the

[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]

a = $.


end of n years,

n =

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AN Consequently, any three of these four quantities being which being a geometrical progression, its sum is found
UTTIES. given, the fourth may be determined.

by the known rules (see ALGEBRA, Div. i.) to be NUITIES.

proper to caution the reader that r does not here signify the rate per cent. per annum, but the interest of 1l. per annum.

and multiplying this by the given annuity a, we shall The method of determining the present value of a similar annuity will readily follow, after what

has been have the amount required, viz. done above; for in this case we must find the present

(1 + value of each year's annuity, as it becomes due. Now the present value of 1l. to be received at the end of a

Hence we readily deduce the following theorems : 1


(1 + r)" - 1 year, is 17r; at the end of 2 years,


= the amount ;
end of 3 years,

&c.; and, generally, at the 1+ 3r

= the annuity;

+ r) 1

log. (1 + 8.r)

– the number of years ; The principles upon which these computations are

log. (1 + r) founded, are illustrated in our treatises on Arithmetic and Algebra

{12 + (n + 1)2}9

= the annual interest; Consequently, the total present value of an annuity

12 + 2 (n + 1) 1 of 11. to continue for n years, is

in which last formula we substitute, for the sake of


1 + 2r 1+ 3r


- 1. which sum, being multiplied by any other annuity a, will be the present value of such an annuity. But as

Note 1. In the above formulæ and investigation we Annuities the summation of this series is very laborious, and as, have supposed the annuity to be payable yearly, and payable half

yearly, after all, it belongs to a case which has little or no consequently n denotes the number of years, and'r the

quarterly, practical application, we shall not detain the reader interest on 11. for one year. But if the annuity &c. upon this subject, but merely give him Simpson's ap- be payable half yearly, or quarterly, or every two or proximation for the same, which may be safely applied three years, we must then consider n to denote the in case such a question should ever occur. Its error is number of payments, and r as the interest payable in excess.

upon ll. for the time of each payment; that is, for Simpson's rule.“ Divide s, or the amount of the half yearly payments, n must be doubled, and r must

n (n 1)

be taken half the annual interest; for quarterly payannuity in the given time, by 1 +nr+


ments, the number of them will be 4 n, and the infor the present value sought."

terest fr: so also for biennial or triennial payments,

the number will be fn, or fin, and the rate 2 r, or 3r; Amount of annuities at compound interest. so that the same formule will apply to any cases of 4. The method of calculating the amount of an

this kind. title of nuities at the end of any given term, improved annually interest upon the annuity to be payable together. It

Note 2. We here suppose the annuity and the at din- has been stated above respecting those at simple in- is obvious, that although an annuity may only be pay

terest; for we have only to find the amount of each able annually, the purchaser may be able to place payment put out at compound interest for the re

his several receipts so that they may improve by half mainder of the term, after it becomes due, and to find yearly or quarterly payments'; if this were taken the sum of all these several amounts.

into consideration, the above formulæ would require

certain modifications; but it would lead us too far If, therefore, s = the amount, -Q = the annuity,

to enter upon this investigation, which, after all, n = the number of years,

is not of very great importance. The reader, howr = the annual interest upon 1l.

ever, will find them treated of in a very luminous

manner by Mr. Baily, in his Doctrine of Annuities. To then, for an annuity of ll. the amount at the end of this work we therefore refer him for the requisite inone year will be 1 + r; and

formation; and we believe the subject has never been 1: (1 + r) :: (1,+ r): (1 + r)

considered under this point of view but by that gentlethe amount for two years; and in the same manner (1 + r)will be the amount for three years, &c.; and, 5. Let us illustrate the above formule by one or generally, for n years, the amount will be (1 + r)". two examples.

Now, as in the case of simple interest, the last pay Required the amount of an annuity of 1001. per annum Illustrated ment of the annuity will have no interest attached to it, for 20 years, at 4 per cent. per annum, and show the by exainand the first will only remain at interest for (n − 1) difference in that amount, on the supposition of yearly,

ples. years; consequently, the whole amount of such an an- half yearly, and quarterly payments. nuity will be expressed by the series

1. For yearly payments we have r = 20, r =:04, 1+ (1 + r) + (1 + r) + (1 + r) +, &c. (1 + r)*-; and a = 100. Whence

[ocr errors][merged small][merged small][ocr errors]


a = 50.

S = 50

S = 25.


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[ocr errors]
[ocr errors]


a = P.

(1 + r)"-1= annuity;

n =

(1 +04) - 1

For 2 years, or payments, we have

= 2,9771, 16 s. 14 d.
S = 100.
• 04


1 ltr:1:: :

for two payments; 2. For half yearly payments, n = 40, r =:02, and

ti (1 + r) Whence


1 (1 +.02) - 1

1 + r : 12:


for three payments; = 3,020 1. 18. 113d.

(1 + i)*(1 + r)' • 02


&c. 3. For quarterly payments, n = 80, r =:01, and



1 +1:1::
a = 25.

for n payments. (1 + r)" , :

(1 + r)"
(1 + .01) - 1
= 3,0411. 15s.

The present worth, therefore, of all the payments, will be •01


If the payments were only made every two years,



(1 + r)'(1 + r)?' (1 + rogol (1 + i)*' then we should have n = 10, r =:08, and a = 200.

1 In this case, therefore, the amount would be

a geometrical progression, of which the ratio is
(1 +.08) - 1
s = 200

= 2,8971. 6s. 3d. and the sum of it, according to the principles delivered •08

in our treatise on Algebra, is
Whence it is obvious, that the oftener the payments


(1 + r)* - 1
are made, the greater will be the value of the whole

1 (1 + r) annuity, as is indeed otherwise obvious.

1 (1 + 7')" As this formula for s, although simple in its form, which being multiplied by any annuity a, will eris somewhat troublesome to put into numbers, tables press its present worth. We obtain thus the following of its several values have been computed, answering theorems, whence any of the four quantities may be to the different values of n and r, for annuities of 11. found when the others are given, viz. from which that for any other proposed annuity may

(1 + r)" - 1 be obtained by simple multiplication. Such is the P = a.

r (1 + r)"

= present worth;
following Table I. for any number of years under 50,
and for all rates of interest from 2 to 7

r (1 + r)"

per per
annum; or from 2 to 7 per cent. per payment.
Let us give an example, by way of illustrating the

co. log. (1 + r)
use of this table.
To what will an annuity of 5001. per annum amount

= number of payments ;

log. (1 + r)
in 40 years, at 4 per cent. per annum, yearly pay-

{12 – (n − 1)9}9

= rate of interest. By the table, the amount of 11. for 40 years,

12 – 2 (n-1) 9 at 4 per cent. is 95.0255 Where co. log. signifies the logarithmic complement

, and
Mult. by

+ 1.
47512.7500 I 19,5121. 15s.
Again, to what will an annuity of 1,000l. per ann.
amount to in 25 years, at 5 per cent. ; the payments

7. The following examples will illustrate these

being made half yearly?
This is obviously the same as an annuity of 5001. for

1. What is the present value of an annuity of 207. E.

per ann. for 40 years, at the rate of 6 per cent per ann. 50 payments, at 21 per cent.

the payments being yearly? By the table, 50 years, at 24 per cent. = 97.4843

Mult. by

Here n = 40, r =:06, and a = 20;

whence p = 20 x

=3001, 183. 6d.

-06 x 1.0640 Present value of annuities at compound interest:

2. What ought to be the annual rent or payment for

554 years, for which a premium of 1001. is paid down, Present 6. The present value of an annuity is such a sum as, allowing interest at 5} per cent. per ann.? value at put out to interest, will enable us to provide for the compound several payments of the annuity as they become due. In Here n= 55), p='100, and r = .055; order to ascertain this sum, we must find the present

055 x (1.055)'s whence a = 100 X

= 51. 16s. value of these several payments ; and the sum of them

(1.055)*5 - 1 will be the total present value sought. Hence, then, let

In this case, as in the former, the theorems for prod p = the present worth, a= the annuity,

ainvolve much arithmetical computation, to avoid which tables of their value are computed to various periods

. n= the number of years,

and for different rates of interest; such are our Table r = the rate of interest per l. per payment.

II. and' III.; the other formula, viz. those for a and s. Now, to find the present worth of 11. for one payment, being by no means so frequently required, it would caly

be a waste of time to reduce them to the tabular form. 1 (1 + r):1::1: present worth of one payment. then n will denote the number of payments, and r the #

Note. When the payments are not made annually

, It


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