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4th = 2 4y2 + y& 5th = 59 5y3 + y& &c. &c. ANIMATED Needle, a needle touched with a magnet. ANNUAL, in Astronomy, any thing which relates to the year, or which returns yearly. ANNUITIES, signify any interest of money, rents, or pensions, payable from time to time, at particular periods. The most general division of anInuities is into annuities certain, and annuities contingent ; the payment of the latter depending upon some contingency; such, in particular, as the continuance of a life. ANNUIties have also been divi. ded into annuities in possession, and annuities in reversion ; the for. Iner meaning such as have commenced, or are to commence inmediately; and the latter, such as will not commence till some particular future event has happened, or till some given period of time luas expired. ANNUITIEs may be farther considered as being payable yearly, half-yearly, or quarterly. The present value of an Annuity is that sum which being improved at compound interest, will be suf. ficient to pay the annuity. The present value of an Annuity certain, payable yearly, and the first payment of which is to be made at the end of a year, is computed as follows: Let the annuity be supposed $100; the present value of the first payment of it, or of a hundred ło to be received a wear ence, is that sum in hand, which being put to interest will amount to £100 in a year. In like man.

ner, the present value of the se.

cond payment, or of £100 to be received two years hence, is that Sun *ich being put to interest

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Substituting therefore as above, a = the annuity, or yearly rent; n = the number of years, or payments; r= the amount of 11. for a year, or for one payment; v = the present value of the annuuty;

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= 3.4021. 16s.
Whence again it follows, that
the greater the number of pay
ments, the greater will be the
amount of the annuity. -
To find the present value of an
annuity by the following table, we
have only to find the amount for
11. at the given rate of interest,
and for the given time; which
multiplied by the given annuity,
or payment, will be the present
worth.
Exam. What is the present value
of an annuity of 40l., per ann. to
continue 20 years, at the rate of 4
per cent. 7
By the table, the amount of
11. for 20 years, at 4 per cent. is
13.590326; therefore
13.590320 x 40 = 6431. 12s. very
nearly.
For what relates to life annui.
ties, see LIFE Annuities and Insu-
7"artices.

TABLE, Showing the present Value of an Annuity of £1.. per Ann. for any

number of Years not exceeding 60, at any rate of Compound
Interest from 3 to 6 per Cent.

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.970874 1.913470 2.82861 I 3,716098 4.579708 5.417.191 6.230.283 7.019692 7.786.109 8.530203 9.252624 9.95 1004 10.63-1955 I 1.296073 I 1 93.7935 12.561 102 13. 1661 18 13.753013 14.323799 14.877-175 15.415024 15.936917 16.443608

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TABLE, Showing the present Value of an Annuity of £1.. per Ann. for any

number of Years not exceeding G0, at any rate of Compound
Interest from 3 to 6 per Cent.

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16.93.55:12 17.413148 17,876842 18.327.031 18.764.108 19. 1884.55 19.600.441 20,000428 20.388765 20.765792 21. 131837 21.487.220 21.832252 22.167235 22.492462 22,808215 23.114772 23.412400 23.70.1359 23.981902 24.254274 24,518713 24.775449 25,024708 25.266707 25,501657 25.7297.64 25.951227 26. 166240 26,374990 20.577660 26.774428 26.965464 27.150936 27.33:1005 27.505831 27.675564

34 perCu.

16.058368 16.481515 16,890.352 17.285364 17.667.019 18.035767 18.392045 18.736.276 19.06SS65 19.390.208 19.700684 20.00066i 20.290494 20,570525 20.841087 21.102500 21.3550.72 21.599.104 21.834882 22.062689 22.2S27.91 22.495450 22.700918 22,899.438 23.091244 23.27.6564 23.455618 23.628616 23.795.765 23.957.260 24.113295 24.264053 24.409713 24.550448 24.686.423 24.817800 24,944734

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15.246963 15.622080 15,982769 16.329580 16.663063 16.933.715 17.292033 17.588494 17.873551 18. 146674 18.411 198 18.66:1613 18.908282 19. 142579 19.367S64 19.5844.85 19.79:27.74 19.993052 20.185627 20.370.795 20.548841 20.720040 20.884652 21.042936 21.1951.31 21.341.472 21.482.185 21.617485 21.747582 21.872675 21.992957 22.108612 22.2198.19 22.326740 22.429567 22.528430 22,623.490

4% perCt. 14.495478 14,828209 15.1466ll 15.451303 15.74287.4 16.021889 16.288.889 16.544391 16.788891 17.02.2862 17.2467.58 17.461012 17.666040 17.862240 18.049.990 18.229656 18.401584 18, 566109 18.723550 18.874210 19.018383 19. 156343 19.288371 19.414709 19.535607 19.651298 19.762008 19.86795() 19.969330 20.066345 20.159181 20.24802? 20.333034 20.414387 20.492236 20.566733 20.633022

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13.7986;2 14 0939.45 14.375.185 14.643034 14.89819.7 15, 141074 15.372451 15.5928; 0. 15.802677 16 002349 16. 192904 16.374.194 16,546352 16.71 i287 16.867893 17.017041 17.159086 17.294368 17.42.3208 17.545912 17.662773 17.774070 17.880066 17.981016 18.077158 18.1687.22 18.255925 18,338977 18.418673 18.493.405 18.1651.46 1S.633.472 18,6985 15 18.760519 18.819542 18.875754 18.92.9290

6 per Ct.

12.550358 12.78.3356 13.003166 13.210534 13.406164

13.5:30721

13.7GHS31

13 9290SG 14,0840.43

14.230230

14.368l4]

14.498246

14,620986 14.736780 14.846019 14.949075 15.046-297

15. 13S016 15.22.4543 15.306.173 15.383.182 15.455832 15.524370 15.5S9028 15.6500.27 15.707572 15.761861 15.813076 15.861 393 15.906974 15.949976 15.990543 16.028S 14 16.064919 16.09S980 16.1311 13 16.161-128

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planet, by which it deviates from the aphelion, or apogee; or it is the angular distance of a planet from the aphelion, or apogee; that is, the angle formed by the line of the apsides, and another drawn through the planet. Kepler distinguishes three kinds of anomaly; mean, eccentric, and true. Mean, or Simple ANoMALY, in the ancient astronomy, is the distance of a planet's mean place from the apogee. But in the modern astronomy, in which a planet is considered as revolving about the sun, in an elliptic orbit, it is the time in which a planet moves from its aphelion to the mean place or point of its orbit. Hence, as the elliptical area is proportional to the time in which the planet describes the arc bounding that area, that area may represent the mean anomaly. Eccentric ANoMALY, or of the Centre, is the arc, intercepted between the apsis, and the point determined by the perpendicular to the line of apsides, drawn through the place of the planet; or it is the angle at the centre of the cirle. True, or Equated ANoMALY, is the angle at the sun, which the planet’s distance from the aphelion appears under; or the angle formed by the radius vector, drawn from the sun to the planet, with the line of the apsides. The finding of the true anomaly, when the mean anomaly is given, is a problem which has engaged the attention of many able astronomers. Dr. Wallis gave the first geometrical solution of it, by means of the protracted cycloid; and Newton did the same at prop. 31. lib. 1. Principia. ANTARCTIC Circle, is a small circle parallel to the equator, at the distance of 23° 28/ from the antarctic or southern pole. ANTARctic Pole, is the southern le of the earth's axis. ANTECEDENT of a Ratio, denotes the first of the two terms of the ratio; thus in the proportion a : b = c : d, a and c are the two antecedents, and b and d the two **. 4.

ANTECEDENTAL Calculus, a branch of analysis invented % J. Glenie, esq. and published by him in 1793. The author professes to employ it, with advantage, instead of fluxions; but it has not been much attended to by other mathematicians. ANTECEDENTIA, a term used by astronomers to denote a planet moving westward, or contrary to the order of the signs. When its motion is eastward, it is said to move in consequentia. ANTILOGARITHMS, the complement of the logarithmic sine, tangent, &c. of an angle; being the difference between them and radius. ANTIPARALLELS, in Geometry, are those lines which make equal angles with two other lines, but in contrary order; that is, calling the former pair the first and second lines, and the latter pair the third and fourth, if the angle made by the first and third lines be equal to the angles made by the second and fourth ; and, on the contrary, the angle made by the first and fourth be equal to à. angles made by the second and third; then each pair of lines are antiparallels to each other; viz. the first and second, and the third and fourth. It has been commonly asserted

of these lines, that each pair cuts

the other into proportional segments, taking them alternately; but this, upon examination, will be found erroneous. ANTIPODES, in Geography, are the inhabitants of two places on the earth diametrically opposite to each other, and who therefore walk feet to feet. It is obvious that antipodes must have the same degree of latitude, but in a different hemisphere; and the difference in longitude is 180°. It is therefore night with one, when it is day with the other; and summer with one, when it is winter with the other. APERTURE, in Hydraulics, is the hole through which a spouting fluid passes. APERTun B, in Optics, is the hole next the object-glass of a telescope, or microscope, ough which the

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