Let us now illustrate these formulæ by a few examples. Required the present worth of an annuity of 100%. per annum for 5 years, at yearly payments, half for the value of an annuity of a yearly payments, and quarterly pounds per year, for n years. In the first case, r = 1.04, n = 5, and a = 100; therefore, by formula I, we have 1.045 -1 100 payments; the rate of interest beIf the annuity is to be at s equaling 4 per cent. in each case. distant times in the year, then r instead of representing the amount of £1 for a year, must be taken for the amount of £1 for the time of the first payment; also n, which is the number of payments in the above, will become ns, and a will a become whence the more gens neral formula will be naity; v = 1.04 1 1.045 £445.18. In the second case, r = 1.02, n 10, and a = 50; therefore v = 1.0210-1 X 50 1.0210 = £449.13. 1.02-1 In the third case, r = 1.01, n = 20, and a 25; therefore 1.0120-1 v= 25 £454.15. X Hence, it is evident that the greater the number of payments, the greater is the present value of the annuity. What annuity, payable yearly, may be purchased for 5 years with 22261.; taking the rate of interest at 4 per cent.? Here we have v 2226, r = m = 1 1 1-0520- 1 1.05-1 X a. 34021. 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 17. at the given rate of interest, and for the given time; which multiplied by the given annuity, or payment, will be the present X 100 33061 12s. worth. Required the amount of the same annuity, at half-yearly payments. Here r 1.025, n = 40, a = 50. - 1 m = m = T 1 1.02540 -1 1.025-1 X a. X50 33701 Os 91d Required the amount of the same EXAM. What is the present value of an annuity of 401. per ann. to continue 20 years, at the rate of 4 per cent.? By the table, the amount of 17. for 20 years, at 4 per cent. is 13.590326; therefore 13.590326 X 40 5437. 12s. very nearly. For what relates to life annuities, see LIFE Annuities and InsuHere r 1 0125, n = 80 a = 25. | rances. at quarterly payments. 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 Yrs. Interest from 3 to 6 per Cent. 3 per Ct. 34 perCt. 4 per Ct. 144 perCt. 15 per Ct. 16 per Ct. 19 13.189682 12.659297 12.15999211 689587 10.827603 12.593294 12.085321 11.158116 22 23 20 14.877475 14.212403 13.590326 13.007936 12.462210 11.469921 11.764077 12.041582 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. Yrs. 3 per Ct, ↑ 34 perct. [ 4 per Ct. | 43 perCt. 5 per Ct. 6 per Ct. 12.550358 12.783356 13.210534 13.764831 13 929086 14.084043 24 16.935542 16.058368 15.246963 14.495478 13.798642 18.16514615.949976 16.064919 return to the same 365d 6h 9m 11s star ANNULAR Eclipse, is an eclipse] but the siderial, ory of the sun, in which the moon appearing less than the sun, leaves a bright ring round the sun's disc. ANOMALISTICAL Year, in Astronomy, called also Periodical Year, is the space of time in which the earth, or other planet, passes through its orbit; which is longer than the tropical year, by reason of the precession of the equinox. For example, the tropical revolution of the sun, with respect to the equinox, is .365d 5h 48m 459 and the anomalis-365 6 15 20 tic revolution is because the sun's apogee advances every year 65" with respect to the equinoxes, and the sun cannot arrive at the apogee till he has passed over the 654 more than the revolution of the year answering to the equinoxes. ANOMALY, in Astronomy, is an irregularity in the motion of a 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 bound ing 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 pole of the earth's axis. ANTECEDENT of a Ratio, denotes the first of the two terms of the ratio; thus in the proportion abc:d, a and c are the two antecedents, and b and d the two consequents. ANTECEDENTAL Calculus, a branch of analysis invented by 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 the 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. APERTURE, in Optics, is the hole next the object-glass of a telescope, or microscope, through which the f light and the image of the object come into the tube, and are thence conveyed to the eye. APEX, the vertex, top, or summit of any thing. as 24-18, for the difference of their squares 24 and 18 is 6, and √6 is to 24 as 1 to 4 or as I to 2. APOTOME Quarta. When the greater term is a rational number, APHELION, or APHELIUM, that and the square root of the differpoint in the orbit of a planet, in ence of the squares of the two which it is at its greatest distance terms has not a rational ratio to it; from the sun: that is, at the ex-as, 4-√√3, where the difference of tremity of the transverse diameter the squares 16 and 3 is 13, and of the elliptic orbit, which is far-13 has not a ratio in numbers thest distant from that focus in to 4. which the sun is placed. APOTOME Quinta, when the less The aphelia of the planets are term is a rational number, and the not fixed; for their mutual actions square root of the difference of the upon each other keep those points squares of the two has not a ra of their orbits in a continual mo- tional ratio to the greater; as √√/6 tion, which is greater or less in-2, where the difference of the the different planets. squares 6 and 4 is 2, and/2 to √6, or 1 to 3, or 1 to /3, is not a rational ratio. APOGEE, in the ancient Astronomy, that point in the orbit of the sun, or a planet, which is farthest distant from the earth; and which corresponds with the aphelion of modern astronomers. It is still used to denote the greatest distance of a body from the earth. The moon, for example, is in its apogee when farthest from the earth; and its aphelion, when farthest from the sun. APOTOME, the remainder or difference between two lines or quantities, which are only commensurable in power. Such as the difference between 1 and 2, or the difference between the side and diagonal of a square. Euclid, in his tenth book, distinguishes these quantities, under six heads, viz. АРОТОМЕ Prima, when the greater term is rational, and the difference of the squares of the two is a square number; as, 35. APOTOME Sexta, where both terms are irrational, and the square root of the difference of their squares has not a rational ratio to the greater; as √6—√√/2, where the difference of the squares 6 and 2 is 4, and 4 to √6, or 2 to 5, is not a rational ratio. APPARATUS, the appendages or utensils belonging to machines; as the apparatus of an air-pump, electrical machine, &c.; meaning the various detached parts which are necessary for putting the ma chinery in action, and for performing experiments, &c. APPARENT, in Mathematics and Astronomy, is used to signify things as they appear to us, in contradistinction from real or true; and in this respect the apparent state of things is often very differ ent from their real state: as is the case of distance, magnitude, &c. APPARENT Conjunction of the Planets, is when a right line, sup. posed to be drawn through their centres, passes through the eye of the spectator, and not through the centre of the earth. APOTOME Secunda. When the less term is rational, and the square root of the difference of the squares of the two terms has to the greater term a ratio expressible in numbers; such is 18-4, APPARENT Diameter of an Obbecause the difference of the ject, is not the real length of that squares 18 and 16 is 2, and 2 is to diameter, but the angle which it 18 as 1 to 9, or as 1 to 3. subtends at the eye, or under APOTOME Tertia. When both which it appears. This angle dithe terms are irrational, and, as in minishes as the distance increases; the second, the square root of the so that a small object at a small difference of their squares has to distance may have the same appa the greater term a rational ratio;rent diameter as a much larger |