78.-Cor. The periodical time in seconds, is 3.1416 V ad's the periodical time in a circle whose radius is d. And, by Art. 63, the periodica time is the same in all circles and ellipses. Fig. 19. R S D 79.-Supposing the centripetal force reciprocally as the square of the distance; to determine the orbit which a body will describe, that is, projected from a given place P, (fig. 19,) with a given velocity, in a given direction PT. By Art. 42, the body will move in a conic section, whose focus is S, the centre of force; and the line of direction PT will be a tangent at the point P. Let the distance SP = d, transverse axis AD = %. f= space a body will descend at P, in a second, by the centri. petal force. v = the velocity the body is projected with from P, or the space it describes in a second. Then v2df is the velocity of a body revolving in a circle, at the distance SP. Then (Art. 44,)v : v2df :: V2-d: V {z. Whence v v Ir= v2dfz — 2ddf, and vvz = 4df2 — 4ddf; and 4df2 — vvz = 4ddf, 4ddf duv whence z = 4df – 4df OD Therefore, if 4df is greater than vv, z is affirmative, and the orbit is an ellipsis: but if lesser, 2 is negative, and the curve is a hyperbola; and if equal, it is a parabola. Draw SR perp. to PT, and let SR = p. Alsu draw from the other focus H, HF perp. to PT. Then (Conics,) the angle SPR = angle HPF, wbence the triangles SPR, HPF, are similar; therefore SP (a): SR (P) :: HP (s—d) : HF = 7 *p; and (Art. 2-d 71,) SR X HF or pp = rectangle DHA or CB', the square 2-d In the triangle SPH, the angle SPH and the sides SP, PH, are given, to find the angle PSH, the position of the transverse axis. 80.- Cor.3. The periodical time in the ellipsis APDB=3.1416X 4df 401-13 2d For 3.1416 periodic time in the circle whose radias is d. And s ad (Art. 59,) qd}: 3.1416 V ? :::*: periodical time in the ellipsis = 3.1416 4ddf ī Xud = 3.1416 X 81.-Cor. 2. The latus rectum of the axis AD is Pitt adj 82.-Cor. 3. Hence the transverse axis and the periodic time will remain the same, whatever be the angle of direction SPT. For no quantities but d, , and e, are concerned; all which are given. SCHOLIUM. --The laws of centripetal and centrifugal forces are the foundation of the theory of astronomy. For it is ascertained, by experience, that the primary planets revolve round the sun in elliptical orbits, the sun being situated in one of the foci. 'he se. condary planets, as the moon, the satellites of Jupiter. &c. also revolve round their primaries in orbits of an elliptical form. The comets are likewise known to revolve round the sun in ellipses, having so great a degree of eccentricity as to approach very nearly to parabolic orbits. The existence of any distant sympathy between matter producing the phenomena of Attraction, Repulsion, and Gravitation, is an appeal to faith founded on the phenomena and ou the difficulty in accounting for them. But, as these sympathies produce force in the bodies, and FORCE is universally the product of matter and motion, Sir Richard Phillips, in his new System of Physics, contends that they are necessarily so many results of imparted motion, and that, instead of giving names to these sympathies, we ought to trace the specific motions which produce the phenomena. Upon this principle he has investigated many phenomena hitherto ascribed to innate powers of matter; and he maintains, that bodies fall towards the centre of the earth, or possess the momentum of weight, owing to the two-fold motions of the earth, of which all the parts of the terrestrial mass are the patients; he also maintains, that the rotation, or motion, of the son, transferred or diffused through the gazeous medium which fills space, carries round the planets in their orbits; and he then descends to the phenomena of Atoms, showing that atomic motion is heat, and gas is atoms in motion; and hence explains the phenomena of Combustion, Animal Heat, Electricity, &e. But, as the general laws of diffused motion, by this system, accord with those of gravitation, as developed by Hooke and Newton, the mathematical results are the same on either hypothesis, though the philosophy and reasoning are very different. In the preceding pages, the general principle of centrifugal and centripetal forces are adopted, whatever be their cause; whether, according to Newton, they consist of an original projectile force impressed at the creation, and a gravitating force arising from innate properties of matter,-or whether, according to the new doctrine, the effect is occasioned by the mutual transfer of motions, wbach create action and reaction between the sun and planets, directly as their quantities, and inversely as the squares of their distances. The results, as govered by the same law, mathematically considered, will be the same. But, for a work of general use, the received language, for many obvious reasons, has been preserved. N.B.—The mean diameter of the Earth being 7960 miles, and its mean distance from the Sun 95,000,000, the diameters and distances of the other Planets may readily be found. As this colume is not intended to supersede a system of Natural Philosophy, the reader who wishes to pursue these subjects further may consult ENFIELD'S Institutes, PLAYFAIR's or Robinson's Elements ; Newton's Principia (of which there is an English edition by MOTTE); VINCE's or SQUIRE's Astrunomy, WOOD's, KIPLING's, or SMITH's Optics ; Hutton's Course on Projectiles und Gunnery; Simson's Essays; and Gregor Y's Mechanics, Hutton's or BARLOW's Mathematical Dictionaries, and Young's Lectures on Natural Philosophy, wili .also supply many desiderata. Readers of the French Language will also find an inexhuustible mine of mathematical research in the Mechanique Celeste of the MARQUIS DE LA Place, and in the works of La Croix and LEGENDRE, of which g more extended notice appears in the Introduction. MISCELLANEOUS QUESTIONS, APPLICABLE TO THE VARIOUS HEADS OF THE PRECEDING VOLUME. N.B. Answers, worked at length, are given in the Key. 1.-A RECKONING of 20 shillings was spent by a company of twenty persons, consisting of officers, sailors, and marines. Each officer paid 2s. 6d., each sailor 12 pence, and cach marine 8 pence. How many persons were there of each denomination ? 2.--A gentleman has a piece of land in the form of an isosceles triangle the perimeter of which is to be 144 chains, to be taken out of a large meadow: Now, bis bargain is, to given ten guineas for every chain of the perpendicular, and fifteen guineas for every chain of the base. He de. sires to know what dimensions he must take, so as to have the most land for the least money ; with the price it will cost bim per acre. 3.-Suppose a weight of 6 oz. is fixed to the end of an inflexible line, void of gravity, 40 inches long; required where another weight of 4 oz. must be fixed to the same line, so that the pendulum may vibrate quickest about the other end. 4.-Given the base and difference of the sides to determine the triangle, when the rectangle of the longest side and difference of the seg ments of the base is equal to the square of the shortest side. 5.-From a given circle to cut off an arc, such that the rectangle under its sine, and the difference of its sine and cosine, may be a maximum. 6.-A staff, equal in length to the depth of a cylinder containing 48 ale gallons, being put dingonally to the bottom, rested with its upper end against the sido, at one foot below the top. Quere, the cylinder's dimensions. 7.- A gentleman, when he first saw his present lady, was four times her age; twelve years after, on their wedding-day, he was only twice her age. 8.- A gentleman sold a borse for 781. by which he gained half as much per cent. as the burse cost him. How much was that? 9.-On a division of the House of Commons, if the number for the motion bad been increased by 40 from the other side, the question would have been carried by 3 to 2; ut, if those against the motion bad received 60 of the other party, the motion would have been defeated by 2 to 1. Query, did the motion sueceed ? and how many members were there in the house? 10.- Investigate a formula for approximating to the mth root of any number; and apply to finding the value of V 161900 to 12 places of decimals? 11. When a parish was inclosed, the allotment of one of the proprietors consisted of two pieces of ground; one of which was in the form of a right-angled triangle; the other was a rectangle, one of the sides of which was egnal to the hypothenuse of the trinogle, the other to half the greater side: but, wishing to bave bis land in one piece, be exchanged bis allotments for a square piece of ground of equal area, one side of which equalled the greater of the sides of the triangle which contained the right angle. By this exchange, he found that he had saved 10 poles of railing. What are the respective areas of the triangle and rectangle; and what is the length of each of their sides? 12. Two men set out at the same time, the one from London to Newcastle, and the other from Newcastle to London. He, from London, walked 10 miles the first day, 16 the second, 22 the third, and so on in arithmetical progression: be, from Newcastle, walked 204 miles every day. When will they meet? 13.-Required five numbers in geometrical progression, so that the sum of the two greatest may be 540, and the sum of the two least 20. 1 1 14.-Given + x+12 = ax; required x. -12-22 1 15.–Given =b; required x. 16.-Of all those parabolic conoids which may be inscribed in tha cone whose altitude is = a, and semi-base = b; query, that whose solidity is a maximum. 17.-In an oblique-angled plane triangle, there is given the difference of the sides which includes the angle of 71° 10', equal to 11, and the line that bisects the said angle is equal to 24; from whence is required a tbeorem that will determine the base and sides of the said triangle. 18.-Suppose a stick, perfectly cylindrical, 50 inches long, and length of the arm 20 inches; query, the centre of percussion, supposing a 2lbs. weight fastened 4 inches from the band, and 5 lbs. quite at the other end. 19.-Let ABCD be a geometrical square, S a spring in an adjacent field, and let there be given the three distances SA = 70, SB = 40, and SC = 60 chains; from thence to find the area and side of the said square. 20.-A rider set out with a certain sum of money. At the first town he came te, be received 480l. He remitted the half of what he then had to bis employer. At the second town he received 4001. and remitted half his casb ; and at the third town he received 1201. and remitted one-third of his cash: he then bad just the sum remaining with which he set out. How much was it? 21.-A, B, C, trade in company with a stock of 2001.; the sum advanced by A, the sum advanced by B, and the gain, being as 3, 5, and 4; and what C put in, when added to the gain, was as much as A and B put in together. What did each advance? 22.-It is required to find the three sides of a right-angled triangle, from the following data :—The number of square feet in the area is equal to the number of feet in the hypothenuse + the sum in the other two sides; and the square described upon the hypotheneuse is less than the square described upon a line equal in length to the two sides, by half the product of the numbers representing the base and area. 23.–Given the base, the perpendicular, and the ratio, of the two sides of a triangle; to find the sides. 24.~Io the equation -=; required z. |