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Annotation.

der to clear up fome paffage, or draw fome conclufion Annota from it.

ANNOTTA. See ANOTTA.

ANNUAL, in a general sense, an appellation given to whatever returns every year, or is always performed within that space of time.

ANNUAL Motion of the Earth. See ASTRONOMY. ANNUAL Leaves, are fuch leaves as come up afresh in the fpring, and perish in winter. These ftand oppofed to Ever-greens.

ANNUAL Plants, called alfo fimply annuals, are fuch as only live their year, i. e. come up in the spring and die again in the autumn; and accordingly are to be recruited every year.

ANNUALRENT is used, in Scots law, to denote an yearly profit due by a debtor in a fum of money to a creditor for the ufe of it

Right of ANNUAL RENT, in Scots law, the original method of burdening lands with an yearly payment for the loan of money, before the taking of intereft for money was allowed by statute.

Annona loured feeds. 3. The fquamofa, or fweet fop, feldom rifes higher than 15 feet, and well furnished with branches on every fide. The leaves have an agreeable fcent when rubbed; the fruit is roundish and fcaly, and when ripe turns of a purple colour, and hath a fweet pulp. 4. The paluftris, or water-apple, grows to the height of 30 or 40 feet. The leaves are oblong, pointed, with fome flender furrows, and have a ftrong fcent when rubbed; the fruit is feldom eaten but by negroes. The tree grows in moift places in all the Weft-India islands. 5. The cherimola, with oblong fcaly fruit, is a native of Peru, where it is much cultivated for the fruit, and grows to be a very large tree well furnished with branches. The leaves are of a bright green colour, and much larger than those of any of the other forts. The fruit is oblong, and scaly on the outfide, of a dark purple colour when ripe, and the flesh is foft and fweet, intermixed with many brown feeds which are smooth and fhining. 6. The Africana, with smooth bluish fruit. 7. The Afiatica, or purple apple. This in fome of the French iflands, as grows alfo in Cuba, in great plenty. The trees rife to the height of 30 feet or more. The fruit is efteemed by the inhabitants of those islands, who frequently give them to fick perfons. 8. The triloba, or North-American annona, called by the inhabitants papaw, is a native of the Bahama islands, and likewife of Virginia and Carolina. The trunks of the trees are feldom bigger than the fmall of a man's leg, and are about 10 or 12 feet high, having a fmooth greenish-brown bark. In March, when the leaves begin to fprout, the bloffoms appear, confifting of fix greenifh-white petals. The fruit grows in clusters of three, and fometimes of four together when ripe, they are yellow, covered with a thin smooth skin, which contains a yellow pulp of a fweet luscious taste. In the middle of this pulp, lie in two rows twelve feeds, divided by as many thin membranes. All parts of the tree have a rank, if not a fetid, fmell; nor is the fruit relished by many except negroes. These trees grow in low fhady fwamps, and in a very fat foil.

Culture. The laft fort will thrive in the open air in Britain, if it is placed in a warm and fheltered fituation; but the plants fhould be trained up in pots, and fheltered in winter for two or three years till they have acquired ftrength. The feeds frequently remain a whole year in the ground; and therefore the earth in the pots ought not to be disturbed, though the plants do not come up the first year. If the pots where thofe plants are fown are plunged into a new hot-bed, they will come up much fooner than those that are expofed to the open air. All the other forts require to be kept in a warm stove, or they will not live in this country.

ANNONÆ PRÆFECTUS, in antiquity, an extraordinary magiftrate, whofe bufinefs it was to prevent a fcarcity of provifion, and to regulate the weight and fineness of bread.

ANNONAY, a small town of France, in the Upper Vivarais, feated on the river Deunre. E. Long. 4. 52. N. Lat. 45. 15.

ANNOT, a small city in the mountains of Provence in France. E. Long. 7. o. N. Lat. 44. 4.

ANNOTATION, in matters of literature, a brief commentary, or remark, upon a book or writing, in or

ANNUEL OF NORWAY, of which mention is made in the acts of parliament of king James III. was an annuel payment of an hundred marks Sterling, which the kings of Scotland were obliged to pay to the kings of Norway, in fatisfaction for fome pretenfions which the latter had to the Scottish kingdom, by virtue of a conveyance made thereof by Malcolm Kenmore, who ufurped the crown after his brother's decease. This annuel was first established in 1266; in confideration whereof the Norwegians renounced all title to the fuc ceffion of the ifles of Scotland. It was paid till the year 1468, when the annuel, with all its arrears, was renounced in the contract of marriage between king James III. and Margaret daughter of Chriftian I. king. of Norway, Denmai, and Sweden.

ANNUITY, a fum of money, payable yearly, half yearly, or quarterly, to continue a certain number of years, for ever, or for life.

An annuity is faid to be an arrear, when it continues unpaid after it falls due. And an annuity is faid to be in reverfion, when the purchaser, upon paying the price, does not immediately enter upon poffeffion; the annuity not commencing till fome time after.

Intereft on annuities may be computed either in the way of fimple or compound intereft. But compound intereft, being found moft equitable, both for buyer and feller, the computation by fimple interest is univerfally disused..

I. Annuities for a certain time.

PROBLEM I. Annuity, rate, and time, given, to find the amount, or fum of yearly payments, and intereft.

RULE. Make 1 the firft term of a geometrical feries, and the amount of 11. for a year the common ratio continue this feries to as many terms as there are years. in the question; and the fum of this feries is the amount of 11. annuity for the given years; which, multiplied by the given annuity, will produce the amount. fought.

EXAMPLE. An annuity of 401. payable yearly, is forborn and unpaid till the end of 5 years; What will then be due, reckoning compound intereft at 5 per cent. on all the payments then in arrear ?

Annuity.

Annuity. 1 5

5

4 5 I: 1.05: 1.1025: 1.157625: 1.21550625? whofe fum is 5.525631251.; and 5.25563125 X 40 = 221.02525 2211. os. 6d. the amount fought.

The amount may alfo be found thus: Multiply the given annuity by the amount of 1 1. for a year; to the product add the given annuity, and the fum is the amount in 2 years; which multiply by the amount of 11. for a year; to the product add the given annuity, and the fum is the amount in 3 years, &c. The former question wrought in this manner follows.

40 am. in 1 year.

1.05

42.00

40

82 am. in 2 years,

1.05

86.10

40

126.1 am. in 3 years.

126.1 am. in 3 years 1.05

132.405 40

172.405 am. in 4 years. 1.05

181.02525

40

221.02525 am. in 5 years. If the given time be years and quarters, find the amount for the whole years, as above; then find the amount of 11. for the given quarters; by which multiply the amount for the whole years; and to the product add fuch a part of the annuity as the given quarters are of a year.

If the given annuity be payable half yearly, or quarterly, find the amount of 11. for half a year or a quarter; by which find the amount for the feveral half-years or quarters, in the fame manner as the amount for the feveral years is found above.

PROB. 2. Annuity, rate, and time given, to find the prefent worth, or fum of money that will purchase the annuity.

RULE. Find the amount of the given annuity by the former problem; and then, by compound intereft, find the prefent worth of this amount, as a fum due at the end of the given time.

EXAMP. What is the prefent worth of an annuity of 401. to continue 5 years, discounting at 5 per cent. compound intereft?

By the former problem, the amount of the given annuity for 5 years, at 5 per cent. is 221.02525; and by compound intereft, the amount of 11. for 5 years, at 5 per cent. is 1.2762815625.

And, 1.2762815625)221.02525000(173.179= 1731. 38. 7d. the prefent worth fought.

The prefent worth may be alfo found thus: By compound intereft, find the prefent worth of each year by itfelf, and the fum of thefe is the prefent worth fought. The former example done in this way follows.

1.2762815625)40.000000000(31.3410
1.21550625)40.0000000 (32.9080
1.157625)40.00000 (34.5535
1.1025)40.000. (36.2811
1.05)40.0

Prefent worth,

(38.0952

173.1788

If the annuity to be purchased be in reverfion, find firft the prefent worth of the annuity, as commencing

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265.94629, prefent worth of the annuity, if it was to commence immediately. 1.05X1.05X1.05=1.157625 L. s. d. 1.157625)265.94629(229.7344=229 14 84/

PROB. 3. Prefent worth, rate and time given, to find the annuity.

RULE. By the preceding problem, find the present worth of 11. annuity for the rate and time given; and then say, As the prefent worth thus found to 11. annuity, fo the present worth given to its annuity; that is, divide the given prefent worth by that of 1 I. annuity. EXAMP. What annuity, to continue 5 years, will 1731. 3s. 7 d. purchase, allowing compound intereft at 5 per cent.?

05:1:1: 201.

1.05X1.05X1.05X1.05X1.05 1.2762815625 1.2762815625)20.00000000 (15.6705.

20

15.6705.

4.3295 prefent worth of 11. annuity. 4.329)173.179(401. annuity. Ans.

II. Annuities for ever, or freehold Eftates.

In freehold cftates, commonly called annuities in feefimple, the things chiefly to be confidered are, 1. The annuity or yearly rent. 2. The price or prefent worth. 3. The rate of intereft. The questions that usually occur on this head will fall under one or other of the following problems.

PROB. I. Annuity and rate of intereft given, to find the price.

As the rate of 11. to 11. fo the rent to the price. EXAMP. The yearly rent of a small eftate is 401.: What is it worth in ready money, computing intereft at 3 per cent.?

As .035: 1:40: 1142.857142=L. 1142 17 14. PROB. 2. Price and rate of interest given, to find the rent or annuity.

As 11. to its rate, fo the price to the rent. EXAMP. A gentleman purchases an eftate for 4000l. and has 4 per cent. for his money: Required the rent? As 1.045 4000: 1: 180l. rent fought. PROB. 3. Price and rent given, to find the rate of intereft.

As the price to the rent, for to the rate. EXAMP. An eftate of 1801. yearly rent is bought for 4000l.: What rate of interest has the purchaser for his money?

As 4000: 180 :: 1:.045 rate fought.

PROB

Annuity.

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PROB. 4. The rate of intereft given, to find how many years purchase an estate is worth.

Divide 1 by the rate, and the quot is the number of years purchafe the estate is worth.

EXAMP. A gentleman is willing to purchase an eftate, provided he can have 2 per cent. for his money: How many years purchafe may he offer?

.025)1.000 (40 years purchase. Ans. PROB. 5. The number of years purchase, at which an estate is bought or fold, given, to find the rate of intereft. Divide 1 by the number of years purchase, and the quot is the rate of interest.

EXAMP. A gentleman gives 40 years purchase for an eftate: What intereft has he for his money?

40)1.000(.025 rate sought.

The computations hitherto are all performed by a fingle divifion or multiplication, and it will scarcely be perceived that the operations are conducted by the rules of compound intereft; but when a reverfion occurs, recourse must be had to tables of annuities on compound interest.

PROB. 6. The rate of intereft, and the rent of a freehold state in reversion,, given, to find the present worth or value of the reverfion.

By Prob 1. find the price or prefent worth of the eftate, as if poffeffion was to commence prefently; and then, by the Tables, find the prefent value of the given annuity, or rent, for the years prior to the commencement; fubtract this value from the former value, and the remainder is the value of the reverfion.

EXAMP. A has the poffeffion of an eftate of 1301. per annum, to continue 20 years; B has the revertion of the fame eftate from that time for ever: What is the value of the estate, what the value of the 20 years poffeffion, and what the value of the reverfion, reckoning compound intereft at 6 per cent.?

By Prob. 1. .06)130.00(2166.6666 value of the eftate. By Tables 1491.0896 val. of the poffeffion.

675.5770 val. of the reverfion. PROB. 7. The price or value of a reverfion, the time prior to the cominencement, and rate of intereft, given, to find the annuity or rent.

By the Tables, find the amount of the price of the reverfion for the years prior to the commencement; and then by Prob. 3. find the annuity which that amount will purchase.

EXAMP. The reverfion of a freehold estate, to commence 20 years hence, is bought for 675.5771. compound intereft being allowed at 6 per cent.: Required the annuity or rent?

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Dr Halley had recourfe to the bills of mortality at Annui Breflaw, the capital of Silefia, as a proper flandard for the other parts of Europe, being a place pretty central, at a distance from the sea, and not much crowded with traffickers or foreigners. He pitches upon 1000 perfons all born in one year, and obferves how many of thefe were alive every year, from their birth to the extinction of the laft, and confequently how many died each year, as in the first of the following tables; which is well adapted to Europe in general. But in the city of London, there is obferved to be a greater disparity in the births and burials than in any other place, owing probably to the vaft refort of people thither, in the way of commerce, from all parts of the known world. Mr Simpson, therefore, in order to have a table particularly fuited to this populous city, pitches upon 1280 perfons all born in the fame year, and records the number remaining alive each year till none were in life.

It may not be improper, however, to obferve, that however perfect tables of this fort may be in themselves, and however well adapted to any particular climate, yet the conclufions deduced from them must always be uncertain, being nothing more than probabilities, or conjectures drawn from the ufual period of human life. And the practice of buying and felling annuities on lives, by rules founded on fuch principles, may be juftly confidered as a fort of lottery or chance-work, in which the parties concerned must often be deceived. But as eftimates and computations of this kind are now be come fashionable, we fhall fubjoin fome brief account of fuch as appear moft material.

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Annuity. Mr Simpfon's Table on the bills of mortality at London.

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From the preceding tables the probability of the continuance or extinction of human life is estimated as follows.

1. The probability that a perfon of a given age fhall live a certain number of years, is measured by the proportion which the number of perfons living at the propofed age has to the difference between the faid number and the number of perfons living at the given age.

Thus, if it be demanded, what chance a perfon of 40 years has to live feven years longer? from 445, the number of perfons living at 40 years of age in Dr Halley's table, fubtract 377, the number of perfons living at 47 years of age, and the remainder 68, is the number of perfons that died during thefe 7 years; and the probability or chance that the perfon in the question Thall live thefe 7 years is as 377 to 68, or nearly as 5% to 1. But, by Mr Simpson's table, the chance is fomething less than that of 4 to 1.

2. If the year to which a perfon of a given age has an equal chance of arriving before he dies, be required, it may be found thus: Find half the number of perfons living at the given age in the tables, and in the column of age you have the year required..

Thus, if the queftion be put with refpect to a per- Annuity. fon of 30 years of age, the number of that age in Dr Halley's table is 531, the half whereof is 265, which is found in the table between 57 and 58 years; fo that a person of 30 years has an equal chance of living between 27 and 28 years longer.

3. By the tables, the premium of infurance upon lives may in fome measure be regulated.

Thus, the chance that a perfon of 25 years has to live another year, is, by Dr Halley's table, as 80 to 1; but the chance that a perfon of 50 years has to live a year longer is only 30 to 1. And, confequently, the premium for infuring the former ought to be to the premium for infuring the latter for one year, as 30 to 80, or as 3 to 8.

PROB. I. To find the value of an annuity of 11. for the life of a fingle perfon of any given age.

Monf. de Moivre, by obferving the decrease of the probabilities of life, as exhibited in the table, compofed an algebraic theorem or canon, for computing the value of an annuity for life; which canon we here lay down by way of

RULE. Find the complement of life; and, by the tables, find the value of 11. annuity for the years denoted by the faid complement; multiply this value by the amount of l. for a year, and divide the product by the complement of life; then fubtract the quot from 1; divide the remainder by the intereft of 11. for a year; and this laft quot will be the value of the annuity fought, or, in other words, the number of years pur chafe the annuity is worth.

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76 4.05 77 3.63 3.57 3.52 3.47 3.41 3.30 78 3.21 3.16 3.11 3.07 3.03 2.95 79 2.78 2.74 2.70 2.67 2.64 2.55 80 2.34 2.31 2.28 2.26

2.23

2.15

The above table shows the value of an annuity of one pound for a fingle life, at all the current rates of intereft; and is esteemed the best table of this kind extant, and preferable to any other of a different conftruction. But yet thofe who fell annuities have generally one and a half or two years more value, than fpecified in the table, from purchasers whose age is 20 years or up

wards.

Annuities of this fort are commonly bought or fold at fo many years purchase; and the value affigned in the table may be fo reckoned. Thus the value of an annuity of one pound for an age of 50 years, at 3 per cent. intereft is 12.51; that is 121. 10s. or twelve and a half years purchase. The marginal figures on the left of the column of age ferve to fhorten the table, and fignify, that the value of an annuity for the age denoted by them, is the fame with the value of an annuity for the age denoted by the numbers before which they ftand. Thus the value of an annuity for the age of 9 and 10 years is the fame; and the value of an annuity for the age of 6 and 14, for the age of 3 and 24, &c. is the fame. The further use of the table will appear in the queftions and problems following..

QUEST. I. A perfon of 50 years would purchafe an annuity for life of 200l.: What ready money ought he to pay, reckoning intereft at 4 per cent. ? L.

By the table the value of 11. is 10.8
Multiply by 200

Value to be paid in ready money 2164.00 Anf. QUEST. 2. A young merchant marries a widow lady of 40 years of age, with a jointure of 300l. a-year, and wants to difpofe of the jointure for ready money: What fum ought he to receive, reckoning interest at 3 per

cent.?

By

Annuity

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