readily follow. This is resolved after various ways, and has been made so plain, that it will be necessary only to refer to a few authors who have treated on this subject. Perhaps it would be injustice to award to any modern writer on Trigonometry, the merit of having given rules entirely his own; for most are satisfied with the presentation of methods that have long been in existence. In point of conciseness, the 8th section of Day's Trigonometry has some advantages although at the expense of important omissions. Hut. ton's method may be found under the article Plane Trigonometry, in the second volume of his mathematics. The subject has been deeply investigated by Halley, Wallis, and numerous other distinguished mathematicians. For formulæ of verification, see Legendre, Lacroix, and Euler. Cagno. li's Trigonometry, a work written in Italian, is valued for entering very extensively into the subject. APPLICATIONS OF MECHANICS. Art. 480. Example 1. See Fig. 186. Since circles are as the squares of their diameters, the circumference of the cylinder AB : the circum. of cylinder CD::12:122::1: 144. 144×500= 72000 the pressure resulting from the force that a man can .7854 × 12.7854=the area of the base of the cyl. inder, which multiplied by 2=1.5708=the number of square feet of water, which multiplied by 62.5 98.175, and this added to 72000=72098.175-the answer. exert. Example 2. By Art. 451, 3000 x 12 x 50-1800000 which divided by 1728=1041.666=the number of solid feet of water, and this multiplied by 62.5=65104.125 pounds. Example 3. 800×6=4800 feet, which multiplied by 3600=17280000 solid feet, and this multiplied by 62.5= 1080,000,000 pounds more than 482142 tons. Example 4. Sin 25° 12::R: the length of the inclination down the dam=28.39441, which multiplied by 1500 feet (500 yards)=42591.6 1500, and this multiplied by 6= 255549.69000 which multiplied by 62.5=15971855.625 pounds. Example 5. See Art. 463. 960-739-221, and 960 divided by 221=4.343=the answer. Example 6. 19.25x8-154.00, and this multiplied by 1000 = 154000.00, which divided by 16=9625 pounds of gold. .24x8=1.92, and 1.92 x 1000-1920.00, which di vided by 16 120 pounds of cork. Example 7. 5049×2.6×1000-13127400.0, which divided by 16-820462, and this divided by 2240=366.277 tons. Example 8. Three fourths of a square acre contains 32670 square feet. Let x=the length of the part under water, then X32670=its solidity, and 32670 × 30=980100=the solidity of the part above water. By Art. 470, 32670x : 980100+32670x::92:1.0263, and x=259.64=the depth of water to which the ice goes. Then 259.64+30=289. 64 the hight of the entire body of ice, and 289.64 × 32670 =9462538.80 its solidity in feet, which multiplied by 1000 =9462538800.00, and this into .92=8705535696 ounces= 544096081 pounds=242900 tons. Example 9. See Art. 463. 1000 1728: 5.346 x, 1000x=9237.888, and x= -9.2378. Example 10. Let x=the quantity of silver added, then 63-x=the gold remaining, and (63—x)÷(10.36)=the number of solid inches of gold, and x÷5.85 the number of sol63-x X id inches of silver: Therefore =8.2245, and 10.36 5.85 x=28.81 ounces. Calculation of the time which Archimides would have required to move the earth with the machine of which he spake to Hiero.* The expression which Archimides made use of to Hiero, king of Sicily, is well known, and particularly to mathematicians. "Give me a fixed point," said the philosopher, " and I will move the earth from its place." This affords matter for a very curious calculation, viz. to determine how much time Archimides would have required to move the earth only one inch, supposing his machine constructed and mathematically perfect; that is to say, without friction, without gravity, and in complete equilibrium. For this purpose, we shall suppose the matter of which the earth is composed to weight 300 pounds the cubic foot; being the mean weight nearly of stones mixed with metalic * Mathematical Recreations. substances, such in all probability as those contained in the bowels of the earth. If the diameter of the earth be 7930 miles, the whole globe will be found to contain 2611074117 65 cubic miles, which make 1423499120882544640000 cubic yards, or 38434476263828705280000 cubic feet; and allowing 300 pounds to each cubic foot, we shall have 1153 0342879148611584000000 for the weight of the earth in pounds. Now we know by the laws of mechanics that, whatever be the construction of a machine, the space passed over by the weight, is to that passed over by the moving power, in the reciprocal ratio of the latter to the former. It is known also, that a man can act with an effort equal only to about 30 pounds for eight or ten hours without intermission, and with a velocity of about 10000 feet per hour. If we suppose the machine of Archimides, then, to be put in motion by means of a crank, and the force continually applied to it is equal to 30 pounds, then with the velocity of 10000 feet per hour, to raise the earth one inch, the moving power must pass over the space of 384344762638287052800000 inches; and if this space be divided by 10000 feet, or 120000 inches, we shall have for the quotient 3202873021985725440, which will be the number of hours required for this motion. But as a year contains 8766 hours, a century will contain 8766 00; and if we divide the above number of hours by the lat ter, the quotient, 36537451768003, will be the number of centuries during which it would be necessary to make the crank of the machine continually than, in order to move the earth only one inch, omitting the fraction of a century. The machine is here supposed to be constantly in action; but if it should be worked only 8 hours each day, the time required would be thrice as long. A father on his death bed bequeathed to his three sons a triangular field, to be equally divided among them; and as there was a well in the field, which must be common to the three co-heirs, and from which the lines of division must necessarily proceed, how is the field to be divided so as to fulfil the intention of the testator? Suppose one line of division EB already drawn. Divide the base AC into three equal paths; and let the points of division be D and G; draw the line ED, and BF parallel to it; then draw the line EF from the point E, and if the point F does not fall without the triangle, the trapezium BEFAB will be one of the thirds required. But if the point F falls without the triangle, the line EA must be drawn to the angle A, and FO parallel to it from the point F, as far as the side AB, which it meets, we shall suppose in O; the line EO will give the triangle BOE, equal to the third of the triangle proposed, BEIČB, the other third may be found in like manner; consequently the remainder of the figure will be a third also. The three lines therefore, EO, EI, and EB, which proceed from the point E, will divide the proposed triangle into three equal parts. A gentleman on his death bed, gave orders in his will that if his lady, who was then pregnant, brought forth a son, he should inherit two thirds of his property, and the widow the other third; but that if she brought forth a daughter, the mother should inherit two thirds, and the daughter one third; the lady however was delivered of two children, a boy and a girl. What was the portion of each? The difficulty in this problem, is to discover in what man. ner the testator would have disposed of his property, had he foreseen that his lady would have been delivered of two children. It has generally been explained in the following manner: As the testaor desired if his wife brought forth a boy, the latter should have two thirds of his property, and the mother the other third; it hence follows, that his inten ! tion was to give the son a portion double that of the mother; and as he gave orders that in case she brought forth a daugh. ter, the mother was to have two thirds of the property, and the daughter the other, there is reason to conclude that he intended the portion of the mother to be double that of the daughter. Consequently, to combine these two conditions, the property must be divided in such a manner, that the son may have twice as much as the mother, and the mother twice as much as the daughter. If we therefore suppose the property to be $30,000, the share of the son will be $171424, that of the mother $85714, and that of the daughter, $42854. I. LATIN TRANSLATIONS FROM 66 THE ANTHOLOGY." Una cum mulo vinum portabat asella, Atque suo graviter sub pondere pressa gemebat. Dic mihi mensuras, sapiens geometer, istas? As the mule and the ass will have equal burthens when the former gives one of his measures to the latter, it is evident that the difference between the measures which they carry is equal to 2. Now if the mule roceives one from the ass, the difference will be 4; but in that case the mule will have double the number of measures that the ass has; consequently the mule will have 8, and the ass 4. If the mule then gives one to the ass, the latter will have 5 and the former 7. II. Aurea mala ferunt Charites, æqualia cuique Mala insunt calatho; Musarum his obvia turba The least number which will answer this problem is 12, for if we suppose that each Muse, the latter would each have three; and there would remain three to each Grace. The numbers 24, 36, 48, &c. will also answer the question; and, after the distribution is, made, each of the Graces and Muses will have 6, or 9, or 12, &c. |