} force int, 3 act il bei act terse or U Also, v=lgt= 2 x 1674 X 12 = may be supposed constant during 396 feet, the last acquired velo- the indefinitely small space of city. uime t, and spaces and velocities EXAM. 2. How long will a body, in uniform motions being proporbe in falling through a space of tional to the times, we shall have 1608 feet these two fundamental propor ,1608 lions, viz. Here s=1608, and t=v 1: or s= { v = 10 seconds nearly. 28 : 0= 111: i EXAM. 3. How far must a body Whence are deduced the following fall to acquire a velocity of 386 formulæ, in which the value of feet per second ? each quantity is expressed in terms 386 of the rest : ö 1. 12 seconds nearly. 2,4 The same law of acceleration 2gs 2. i=2g i 3. š= vi in that case vary as the sines of 29 the angles of inclination of the i planes, that is, the force down the 4. • • • • inclined plane is to the whole 2g = 1 force of gravity as the sine of the angle of inclination of the plane These formulæ, like those in the to radius. If, therefore, the angle preceding part of this article, are of inclination of the plane be a, equally applicable to the destructhe force down the plane will be tion of motion and velocity, by sin, a Xg; and by using this in- means of retarding forces, as to stead of g, the above formulæ will the generation of them by means be equally applicable to the de- of accelerating forces. scents of heavy bodies down in The motion of a body ascending, clined planes. or impelled upwards, is diminished Of variably accelerated motion. or retarded from the same prinHaving illustrated the laws of ac- siple acting in a contrary direction, celerated motion, when the acce in the same manner as a falling lerating forces are constant, and body is accelerated. deduced the formulæ for expres A body projected upwards, rises sing them in final determinate till it has lost all its motion; which quantities, we subjoin those that it does in the same time that a fallbelong to the cases of variably ing body would have acquired a accelerated motions. velocity equal to that with which Here the form:læ will be fiuxio- the body was thrown up. Hence nary, expressions, the fluents of the same body projected up, will which, adapted to particular cases, rise to the same height, from will give the relation of time, which, if it fell, it would have acspace, velocity, &c. Let t denoté quired the velocity with which it the time of motion; v the velocity was projected upwards. And hence generated by any force; s the the heights to which bodies thrown Space passed over; and 2g the upwards with different velocities, variable force at any part of the ascend, are to each other as the motion, or the velocity that the squares of those velocities. force would generate in one second ACCELERATION, in Astronomy, of time, if it should continue inval is a term applied to the fixed stars. riably like the force of gravity The diurnal acceleration is the during that one second, and the time by which the stars, in one value of this veiocity, 2g, will be diarnal revolution, anticipate the in proportion to 324 feet, as that mean diurnal revolution of the sun, variable force is to the force of which is 31 557%" of mean time, gravity. Then, because the force' or nearly 31 561: ; that is to say, a acqui : the star rises or sets, or passes the me- quantities. The general principle ridian, 31 56" sooner each day. upon which it is founded, is that This apparent acceleration of the the whole is equal to all the parls. stars is owing to the real retarda. Addition is indicated or expresstion of the sun, which depends ed by writing the sign +, which is upon his apparent motion towards called the sign of addition between the east, at the rate of about 591 the numbers or quantities; thus 8ļ" of á degree every day. In 7+5, and a+b, indicate the addition consequence of this, the star which of 7 and 5, and of a and b. The result of the addition is called passed the meridian at the same inoment with the sun yesterday, is the sum. It may be expressed in about 591 8\" beyond the meridian one number, when the proposed numbers are all of the same kind, to the west, when the sun arrives and when I in any of them is equal at it; and this distance will require to 1 in each of the others. When about 31 56" of time for the sun to this is not the case, an alteration pass over, and therefore the star nust be made in the numbers, and will anticipate the motion of the if such an alteration cannot be sun at this rate every day. ACCELERATION of a Planet, tained in one number. In Algebra, made, then the sum cannot be obA planet is said to be accelerated in its motion, when its real diurnal quantities cannot be collected into motion exceeds its mean diurnal the same power of the same letters. one quantity, unless they consist of motion ; and retarded in its motion, when the mean exceeds the reai ted, according to the nature of the Addition is variously denomina. diurnal inotion. arises from the change in the dis numbers or quantities that are to be added. Thus, tance of the planet from the sun, which is continually varying; the the sum of several simple or ab Simple Addition is the finding of planet moving always quicker in perits orbit when nearest the sun, and stract numbers; and it is thus formed, slower when furthest off. ACCELERATION of the Moon, places stand under each other. Arrange the numbers so that like is the increase of the moon's mean Collect the sum of the units, write motion from the sun, compared below the column what is over with the diurnal motion of the earth ; and it appears that, from a in the next column. Do this with tens, and for every ten reckon one certain cause, it is now a little | all the columns, in order, to the left quicker than it was formerly. hand. La Place has shown this acceleration of the moon's motion to arise 6973 69321 from the action of the sun upon 4218 1168 the moon, combined with the vari 9374 ation of the eccentricity of the 8231 12345 earth's orbit. By the present diminution of the eccentricity, the 28796 Sunis 83734 moon's mean motion is accelerated; but when the eccentricity is arrived at the minimum, the acce Addition may be proved in varileration will cease; after this the ous ways: One may begin altereccentricity will increase, and the nately at the top and bottom of moon's mean motion will be re- the columns, and see if the results tarded. correspond. The excess over 9 ACUTE, or SHARP, a terni op- may be noted in each line, and posed to obtuse ; thus we say, acute- also in the sum, and if the sum of angled triangle, ucute-angled cone, the former be equal to the latter, &c. or ditfer from it only by one or ADDITION, one of the funda. more 9's, there can be no error but mental operations in Arithmetic 9 in the addition : Thus in the first and Algebra, denotes the finding of the preceding examples, the reof one number or quantity, equal mainder on the lines are 7, 6, 5, 5; to two or more given numbers or the remainder in the sum 3; and EXAMPLES. 600 + $. qr. b. 47 zre of the at are the sum of the former (23) abating 63 35 189 210 two nines, leaves 5 also. Hence 3+ + 4 2 13 J2 the addition is right. 407 Compound Addition is the col = 33s. lld. 11. 13s. 11d. lecting several quantities of differ 12 ent denominations into one sum. Place the same denominations ADDITION of Decimals, is tinding the sum of several nunibers, conto stand directly under each other. Add up the figures in the lowest sisting partly of integers and partly denomination, and find how many of uecimals, or of decimals only. units of the next higher denomi. Arrange all the quantities so that nation are contained in their sum. the several decimal points may Write down the remainder, and fall in a line directly under each carry these units to the next higher other, and then proceed as in simdenomination; which add up in ple addition. EXAMPLES. 1. Find the sum of 34.17 ; 19. 143; 167.13; and 143.5. EXAMPLES 2. Find the sum of .1176; .1314; 1. d. cwl. .746; and .1468. 11 3 17 2d. Exam. 6 2 34.17 1176 19.143 .1344 167.13 .746 143.5 .1468 106 7 104 Sums 155 3 24 363.943 Sums 1.1418 ADDITION of Fractions is the ad. ADDITION of Intermediate Deci. ding of several fractions into one mals, is the finding the sum of any number of circulating decimals. 1. If the fractions are of differ 1. Reduce all the decimals to ent denominations, they must be first reduced to the same denomi- their equivalent fractions; and the sum of these will be the answer nation. 2. Reduce all mixed num required; or, bers to improper fractions; and all 2. Carry on the repetends till fractions, having different denomi. nators , to the same denominator. they all begin and end their periods of circulation in the same lines, 3. Add all the numerator's into one and let the circulation of each be sum; which, placed over the com- carried two figures beyond this mon denominator, will be the an- place; then add them up as in the swer required. former rule, observing not to set EXAMPLES. down any thing in the first two 1. Add together the fraction 3, ber from them to the conterminous places; only carry the proper non period; and the result will give 35 14 10 the true period of circulation in First, 70; }= 7034 70 the sum required. EXAMPLE Add 3.8; 78.3476 ; 42.84 ; and 15.5 = 3.6666666 66 pound. 63 78.3476 = 78.3476476 47 4 of a guinea= of a shilling, 42.84878-18 48 35 15,5 15 500000000 of a pound 2 of a shilling; Therefore, The suni ... Sun. that like 7 other ts, write is over kon obe his with the left }, and 59 n vari alter. om of esults ver 9 and am of atler, ne or r but first le re5, 5; 42.84 140.3627991 5a + 33y 33y + In this example, the periods of Fractions, reduce to a common decirculation do not commence to nominator, and then add the uu. gether till after the seventh place merators for the sum required. of decimals; they are then carried on two places further, in order to EXAMPLES. ascertain what ought to be carried 3a2 50 to the conterminous period; which 1. Add together 662 46 m the present case is 1, as will ap 1265 pear from the above operation. 5a 10ab 3a29ab2 50 50 Note. There may arise cases in 662 12639 46 1203 » 1203 1213 which it will be necessary to carry 10ab 9ab2 5c 10ab+9a2b?+5c the circulation on to three or more + + 1264 72 4x 21.2 44.2 65r ADDITION, in Algebra, is finding + + lly Зу the sum of several algebraical 33y quantities, and connecting those 3. And, quantities together with their 7x + 4y 5.1-3y 7cx + 4cy proper signs. And this is generally + 3abc divided into the following cases. Case 1. When all the indetermi 15x-9y 76x + 15x + 4cy-9y nate letters are the same, and have 3abc 3abc the same sign. 170 + 15) x + (4c-9) y Add the co-efficients of the several quantities logether, and prefix 3 ab: before the sum the proper sign, In the addition of Surda, reduce whether it be plus or minus. all the given quantities to their EXAMPLES. most simple form ; then add the co-efficients of the radicals which 7a + 36 + 5.3 13y have the same index and the same + 5a + 46 + 4x ily number. + ba + 116 + 142 EXAMPLES. Thus, + 18a + 18b Sums + 23% 3ly ✓8+18=2/2+3 /2=512 712 +27=2/3+33=5V3 Case 2. When the quantities are 1080++ 32a=3a 34a+2 3440 the same, but have different signs. Add all the like quantities toge = (32+2 3 4a. ther that have also the same signi, Note. When the quantities are and thus two separate suns will reduced to their lowest terms, or be obtained ; then subtract the less simplest forn, and have different of these from the greater, and pre-indices and numbers, they can fix before the remained the sign of only be added together by means the greater sum. of the sign + placed between them. EXAMPLES. Thus, v 18 + 108 = 3 V2+6/3, - 4xy + 3x3 cannot be reduced to a simpler bab + 702 + 6xy - 11:3 form than that above; and the -- 13c2 + 4xy – 1423 same with various others. ADDITION of Ratios, is the same 9c2 Sums + 6xy - 2223 | as composition of ratios; thus, if a :b = c;d; Note 1. When the leading quan- then by addition, or composition, tity of any algebraical expression a+b:a=c+d:c has no sign, it is supposed to be or, a +b:b=c+d:d. affected with the sign +. ADDITIVE, denotes something Note 2. Unlike quantities can to be added to another, in contra. only be added by means of the distinction to something to be taken sign + placed between them. away, or subtracted. Thus, astroIn the addition of Algebraic nomers speak of additive equa10 7y +7ab 30% + 4ab + 5ab 1 E ed. 337 bc reduce ther od the which e Sales tions; and geometricians, of addi.f exposed to the fire, produces a tive ratios, &c. &c. vehenient blast ot wind. ADFECTED `Equation, in Alge ÆOLUS'S Harp, an instrument bra, is that in which the unknown so naned from its producing agreequantity is found in two or inore able harmony, merely by the acdifferent degrees or powers; thus, tion of the wind. It is thus con28 — px? + ax=a, is an ad fected structed : let a box be made of as equation, having three different thin deal as possible, of the exact powers of the unknown quantity x length answering to the width of entering into its composition. Such the window in which it is intended equations are distinguished from to be placed, five or six inches simple, which involve but one deep, and seven or eight inches power. wide; let there be glued upon it ADHESION, in Philosophy, is a two pieces of wainscot, about half species of attraction which takes an inch high and a quarter of an place between the surfaces of inch thick, to serve as bridges for bodies, whether similar or dissimi- the strings ; and within side, at lar, and which, in a certain degree, each end, glue two pieces of beech, connects them together; differs about an inch square, of length from cohesion, which, uniting par- equal to the width of the box, ticle to particle, retains together which are to sustain the pegs ; into the component parts of the same these fix as many pins, such as are mass. The power of adhesion is used in a harpsichord, as there are proportional to the number of to be strings in the instrument, touching points, which depends half at one end and half at the upon the figures of the particles other, at equal distances : it now that form the bodies; and in solid remains to string it with small catbodies, upon the degree in which gut, or blue first fiddle-strings, fixtheir surfaces are polished and ing one end to a small brass pin, compressed. The effects of this and twisting the other round the power are extremely curious, and opposite pin. When these strings many instances astonishing. are tuned in unison, and the inMusschenbroek relates that two strument placed with the strings cylinders of glass, whose diameters outward in the window to which were not quite two inches, being it is fitted, it will, provided the air heated to the same degree as boil. blows on that window, give a sound ing water, and joined together by like a distant choir, increasing or means of melted tallow lightly put decreasing according to the between them, adhered with a force strength of the wind. equal to 130 pounds: lead, of the ÆRA, in Chronology, is the same same diameter and in siinilar cir- as Epoch, and means a fixed point cumstances, adhered with a force of time, from which to begin a of 275 pounds; and soft iron with compulation of the succeeding one of 300 pounds. Grains. years. Gold adheres to mercury, ÆRA also means the way or with a force of 446 mode of accounting time. Thus, Silver 429 we say, such a year of the Chris Tin 418 tian æru, &c. Lead 397 Christian Æra. It is generally Bismuth 372 allowed by chronologers, that the Platina 282 computation of time from the birth Zinc 204 of Christ, was only introduced in Copper 242 the sixth century, in the reign of Antimony 126 Justinian; and is generally as. Iron 115 cribed to Dionysius Exiguus. See Cobalt 8 Epoch. ÆOLIPILE, ÆOLIPILA, an in ÆRIAL, Perspective, is that strument consisting of a bollow which represents bodies diminishmetalline ball, with a slendered and weakened in proportion to neck or pipe arising from it. This their distance from the eye; but it being filled with water, and thus relates principally to the colours in -5/? -5/3 244 s are 15, or erent са jeans Ween 5/3, pler the same . on, sing tra ken troua. |