PROBLEM II. To find the dominical letter for any year, according to the Gregorian or NEW STYLE. RULE. Divide the centuries by 4, and take the remainder from 3. Add twice this remainder to § of the odd years, and divide the sum by 7. If there is no remainder, the domi nical letter is G; if 1 remainder, F, &c., as in the preceding rule. What is the dominical letter for 1895 ? 4)18 cent. 2 3 2= 1 2 2 x 1 Odd years 95 23 7)120 17+1 Dividing 18 centuries by 4, there is 2 re. mainder. Taking this remainder from 3, wo have a remainder of 1. Twice 1 added to 95 years plus of 95, (rejecting the fraction,) gives 120, which divided by 7 gives a re. mainder 1, indicating that the dominical letter is the 1st below G, which is F. 8. Find the dominical letter for 1833. 9. Find the dominical letters for 1856. 10. Find the dominical letters for 2040. 11. Find the dominical letter for 1911. 12. Find the dominical letter for 1799. 13. Find the dominical letters for 1876. 14. Find the dominical letter for 1921. PROBLEM III. To find the day of the week corresponding to any given day of the month. RULE. The dominical letter found by one of the preceding rules, will indicate the day on which the first Sunday in January will fall. The day of the week for the corresponding day of each succeeding month, may be found by the initials of the following couplet: At Dover Dwells George Brown Esquire, On what day of the week was the Declaration of Independence signed ? The dominical letters for 1776 were G, F. Therefore the first Sunday in January was the 7th of the month. Then A representing the 7th Jan., D would represent the 7th Feb.; D the 7th March; Ğ the 7th April; B the 7th May; E the 7th June; and G the 7th July. But 1776 being a Leap Year, the dominical letter after February is one day earlier in the month, and a day of the month which would otherwise be represented by G, will be represented by A or Sunday. The 7th July, therefore, came on Sunday, and the 4th on Thursday. The initials O. S. denote the Old Style. In all cases not thus marked, the New Style is understood. 15. Washington was born on the 22d Feb. 1732. was the day of the week? What 16. The pilgrims landed at Plymouth, Dec. 11, 1620, O. S. What was the day of the week? 17. The Battle of Waterloo was fought June 18, 1815. Is it probable that a letter, purporting to have been written at the time, and dated Friday, June 18, is authentic? 18. On what day of the week was Oct. 11, 1492, O. S., the day that Columbus discovered America? 19. On what day of the week did Columbus set sail, Aug. 3, 1492, O. S. 20. On what day of the week will a note, at 90 days, dated Aug. 20, 1844, become due, allowing 4 days grace? 21. On what day of the week will a note, at 60 days, dated May 27, 1844, become due, allowing 3 days grace? 22. Washington died on the 14th day of the last month of the last year of the last century. What was the day of the week? 23. On what day of the week will be the 3d of April, 1896? MENSURATION. PROBLEM I. To find the area of any surface bounded by four sides, the opposite sides being equal. RULE. Multiply one of the sides by the perpendicular let fall upon it, from the opposite side. 1. The length of an oblong rectangular field is 40 rods, and the breadth 16 rods. How many square rods does it contain? How many acres? 2. What are the contents of a four-sided field, whose opposite sides are equal, the length being 81 rods, and the distance between the longest sides, 13 rods? 3. The average length of Pennsylvania is about 300 miles, and the breadth 157 miles. What is the area? 4. What is the area of a rhombus, the side being 73.5 rods, and the breadth 61.25 rods? 5. How many square feet in a board that is 14 ft. long, and 11 inches wide? 6. How many square yards in a floor 14.3 ft. long, and 10 ft. wide? PROBLEM II. To find the area of a trapezoid, or figure of four sides, two of which are parallel. RULE. Multiply the sum of the two parallel sides by half the distance between them. 7. The two parallel sides of a trapezoid measure 11 and 15 inches respectively, and the height is 8 inches. What is the area? 8. What is the area of a field, two sides of which are parallel, and measure 72.5, and 89.25 rods, the distance between them being 39 rods? 9. What is the area of a trapezoid, one of the parallel sides measuring 96 rods, the other 634 rods, and the distance between them being 84.6 rods? PROBLEM III. To find the area of a triangle. RULE. Multiply one of the sides by one half of the perpendicular let fall from the opposite angle. 10. How many acres in a triangular meadow, one side measuring 1273 rods, and the perpendicular 82.41 rods? 11. The base of a triangular lot is 96 rods, and the perpendicular distance from the opposite angle is 35 rods. What is the area? 12. What is the area of a triangle whose base is 11 inches, and perpendicular, 28.49 inches? Any surface bounded by straight lines, may be divided into triangles, and the area of each triangle obtained. The sum of the several areas is the area of the whole surface. PROBLEM IV. To find the area of a circle. RULE. Multiply half the diameter by half the circumference, or multiply the square of the diameter by .785398. 13. The diameter of a circle is 363 feet. What is the area? 14. What is the area of a circle whose diameter is 97 miles? PROBLEM V. To find the area of an ellipse, the two diameters being given. RULE. Multiply the longer by the shorter diameter, and the product by .785398. 15. A house lot in the form of an ellipse has one diameter 110 feet, and the other 45 feet. What is the area? 16. What is the area of an ellipse, whose diameters are 25 and 17.5 feet? PROBLEM VI. To find the surface of a sphere. RULE. Multiply the square of the diameter by 3.1415926; or, multiply the diameter by the circumference. 17. What is the area of the earth's surface? 18. The circumference of a globe is 252 inches. What is the area? 19. The diameter of a globe is 11.5 inches. the area? What is PROBLEM VII. To find the area of the convex surface of a cylinder. RULE. Multiply the circumference of the base by the height of the cylinder. 20. The diameter of a cylindrical column is 6 feet, and the height 60 feet. What is the area of the convex surface? 21. What is the area of the whole surface of a cylinder, whose diameter is 7.5 feet, and height 49 feet? PROBLEM VIII. To find the solid contents of a cylinder. RULE. Multiply the area of the base by the height. 22. The diameter of a cylinder is 13 inches, and the height 69 inches. What are the solid contents? PROBLEM IX. To find the solid contents of any cylindrical body, whose sides taper uniformly,* as the trunk of a tree. * Such a body is called the frustrum of a cone. |