equator where the earth's circumference measures 24873. 12 English miles; and at what rate per hour are the inhabitants of London carried in the same direction, where a degree of longitude measures 42.99 miles. FIRST.-For the Inhabitants at the Equator. 23 hours 56 minutes are equal to 23.9333 hours.-Now, 360 degrees multiplied by 42. 99 miles, give 15476.4 miles ;-And, As 23.9333: 15476.4 : 1 646 miles. 15476.4000 .1116420 957332 .1590880 1435998 .154882 Hence, the inhabitants under the equator are carried at the rate of 1039 miles every hour, and those of London 646 miles per hour, by the earth's motion round its axis. Example 3. If a ship sails at the rate of 114 knots per hour; in what time would she circumnavigate the globe, the circumference of which is 24873. 12 miles? 11 knots are equal to 11. 25 miles.-Now, As 11.25 : 1 :: 24873.12: 2210.9 hours. 2250 .2373 2250 .1231 1125 .10620 10125 ..495 Hence, the required time is 2210.9 hours; or 92 days, 2 hours, and 54 minutes, PROPORTION, AND PROPERTIES OF NUMBERS. If three quantities be proportional, the product or rectangle of the two extremes will be equal to the square of the mean. If four quantities be proportional, the product of the two extremes will be equal to the rectangle or product of the two means. Thus, Let 2.4. 8. 16 be the four quantities; then, the rectangle of the extremes, viz. 16 × 2, is equal to the rectangle of the means, viz. 4 × 8, or 32. If the product of any two quantities be equal to the product of two others, the four quantities may be turned into a proportion by making the terms of one product the means, and the terms of the other product the extremes. Thus, Let the terms of two products be 10 and 6, and 15 and 4, each of which is equal to 60; then, As 10: 4::15: 6. As 4: 6::10: 15. As 6: 15:: 4: 10, &c. &c. If four quantities be proportional, they shall also be proportional when taken inversely and alternately. If four quantities be proportional, the sum, or difference, of the first and second will be to the second, as the sum, or difference of the third and fourth is to the fourth. Thus, let 2.4.8. 16 be the four proportional quantities; then As 2+ 4:4:8+16: 16; or, as 42:4:: 168: 16. If from the sum of any two quantities either quantity be taken, the remainder will be the other quantity. If the difference of any two quantities be added to the less, the sum will be the greater quantity; or if subtracted from the greater, the remainder will be the less quantity. If half the difference of any two quantities be added to half their sum, the total will give the greater quantity; or if subtracted, the remainder will be the less quantity. If the product of any two quantities be divided by either quantity, the quotient will be the other quantity. If the quotient of any two quantities be multiplied by the less, the product will be the greater quantity. The rectangle or product of the sum and difference of any two quantities, is equal to the difference of their squares.-Thus, = Let 4 and 10 be the two quantities; then 4+10=14; 10-46, and 14x6 84.-Now, 10 x 10 = 100; 4x4 16, and 100-16 84. = The difference of the squares of the sum and difference of any two quantities, is equal to four times the rectangle of those quantities.-Thus, = Let 10 and 6 be the two quantities; then 10+6 = 16 × 16 = 256;10-64×4 16.-Now, 256-16= 240; and 10×6×4 = 240. The sum of the squares of the sum and difference of any two quantities, is equal to twice the sum of their squares. Thus, = 10+6= 16× 16 256; and 10-6=4x4-16; then 256+16=272. Again, 10×10 = 100; 6×6=36, and 100 +36 136×2 = 272. = If the sum and difference of any two numbers be added together, the total will be twice the greater number. Thus, 10+6= 16; and 10-6= 4; then 16+4= 20; and 10 x 2 = 20. If the difference of any two numbers be subtracted from their sum, the remainder will be twice the less number. Thus, 10-64; and 10+6= 16; then 16-4 12;-and 6×2 = 12. = The square of the sum of any two numbers is equal to the sum of their squares, together with twice their rectangle. Thus, 10+6= 16; and 16 × 16 = 256. Again, 10 x 10 = 100; 6×6= 36, and 100+36 136; then, 10×6×2 = 120; and 120+136 = 256. = The sum, or difference, of any two numbers will measure the sum, or difference, of the cubes of the same numbers; that is, the sum will measure the sum, and the difference the difference. The difference of any two numbers will measure the difference of the squares of those numbers. The sum of any two numbers differing by an unit (1,) is equal to the difference of the squares of those numbers. Thus, 9+8= 17; and 9 x9= 81; 8x8 = 64; now, 81-64 = 17. If the sum of any two numbers be multiplied by each number respect ively, the sum of the two rectangles will be equal to the square of the sum of those numbers. Thus, 10+6= 16; now, 16x 10 = 160; 16×6=96; and 160+96 = 256. Again, 10+6= 16; and 16x16 = 256. The square of the sum of any two numbers is equal to four times the square of half their sum.-Thus, = 10+6= 16; and 16 x 16 = 256; then 10+6 16+2 S, and 8x8 x4 = 256. of The sum of the squares of any two numbers is equal to the square their difference, together with twice the rectangle of those numbers.Thus, 10×10 = 100; 6×6=36; and 100+36 136.-Again, 10-64; and 4x4 = 16; 10×6×2 = 120; and 120+16 = 136. The numbers 3, 4 and 5, or their multiples 6, 8 and 10, &c. &c., will express the three sides of a right angled plane triangle. The sum of any two square numbers whatever, their difference, and twice the product of their roots, will also express the three sides of a right angled plane triangle. Thus, = Let 9 and 49 be the two square numbers :-then 9+49= 58; 49—9 = 40.-Now, the root of 9 is 3, and that of 49 is 7;-then 7x3x2 = 42: hence the three sides of the right angled plane triangle will be 58, 40, and 42. The sum of the squares of the base and perpendicular of a right angled plane triangle, is equal to the square of the hypothenuse. The difference of the squares of the hypothenuse and one leg of a right angled plane triangle, is equal to the square of the other leg. The rectangle or product of the sum and difference of the hypothenuse and one leg of a right angled plane triangle, is equal to the square of the other leg. The cube of any number divided by 6 will leave the same remainder as the number itself when divided by 6.-The difference between any number and its cube will divide by 6, and leave no remainder. Any even square number will divide by 4, and leave no remainder; but an uneven square number divided by 4 will leave 1 for a remainder. PLANE TRIGONOMETRY. The Resolution of the different Problems, or Cases, in Plane Trigonometry, by Logarithms. ALTHOUGH it is not the author's intention (as has been already observed,) to enter into the elementary parts of the sciences on which he may have occasion to touch in elucidating a few of the many important purposes to which these Tables may be applied; yet, since this work may, probably, fall into the hands of persons not very conversant with trigonometrical subjects, he therefore thinks it right briefly to set forth such definitions, &c. as appear to be indispensably necessary towards giving such persons some little insight into this particular department of science. PLANE TRIGONOMETRY is that branch of the mathematics which teaches how to find the measures of the unknown sides and angles of plane triangles from some that are already known. It is divided into two parts; right angled and oblique angled :-in the former case one of the angles is a right angle, or 90; in the latter they are all oblique. Every plane triangle consists of six parts; viz., three sides and three angles; any three of which being given (except the three angles), the other three may be readily found by logarithmical calculation. In every triangle the greatest side is opposite to the greatest angle; and, vice versa, the greatest angle opposite to the greatest side.-But, equal sides are subtended by equal angles, and conversely. The three angles of every plane triangle are, together, equal to two right angles, or 180 degrees. If one angle of a plane triangle be obtuse, or more than 90°, the other two are acute, or each less than that quantity: and if one angle be right, or 90°, the other two taken together, make 90: :-hence, if one of the angles of a right angled triangle be known, the other is found by subtracting the known one from 90.-If one angle of any plane triangle be known, the sum of the other two is found by subtracting that which is given from 180; and if two of the angles be known, the third is found by subtracting their sum from 180° The complement of an angle is what it wants of 90°; and the supplement of an angle is what it wants of 180: In every right angled triangle, the side subtending the right angle is called the hypothenuse; the lower or horizontal side is called the base, and that which stands upright, the perpendicular. |