1 CO2 = 9 DO 6.25 (83, corol.) DC2 = 275 and DC = 1.6583 CT 3 Whence the content of the segment PTO = 14.437 hemisphere WTZ = 56·549 zone WPOW = 42.112 diff. Ans. 3. If a segment 3 inches high be cut from a globe 9 inches in diameter, what is its cubic content? Ans. 98.96 inches. 4. Suppose the muzzle of a 32 pounder is stopt with a 42lb. ball; required the content of the part within the bore, if of an inch has been allowed for windage ? Ans. 365 cubic inches. We recommend the use of models for all the solids having plane sides. The planes may be cut in stiff paste-board; and when folded up, the edges are easily fastened together with slips of thin paper and gum-water. ADDITIONAL EXAMPLES IN PRACTICAL GEOMETRY, TRIGONOMETRY, and MENSURATION. 1. If the diagonal of a square redoubt he 67 yards; what is the length of the side? Ans. 47.376 &c. yards. 2. The sides of three squares being 4, how long is the side of that square which 5, and 6 feet; then equal to all three ? Ans. 8.7749 feet, nearly. 3. If the lengths of two lines are 20 and 30 inches; what is the length of that line which is a geometrical mean between them? Ans. 24.4949 in. nearly 4. If the diameter of a circle be 50 yards; what is the length of a chord which is 5 yards distant from the centre ? Ans. 48.9899 yds. 5. If a point be 20 inches distant from a circle whose diameter is 20 inches, and a line 30 inches long be drawn from that point to the circumference; what is the length of that part of the line which is without the circle? Ans. 26 inches. 6. Suppose in the last example, the line is drawn from the given point to make the intercepted chord 10 inches; what is the length of the part without the circle? Ans. 23.7228 &c. inches. VOL. I. 3 с 7. In the preceding example, what is the length of the tangent to the circle drawn from the given point? Ans. 28.284 &c. in. 8. To what extent on the surface of the sea (exclusive of the effect of refraction) can a person see from the top-mast-head of a man of war, his height above the water being 30 yards, and the earth's diameter 7960 miles? Ans. 11.6 miles, nearly. 9. If a line 10 inches long be cut according to mean and extreme proportion; what are the lengths of the two parts? Ans. 6.18 and 3.82 in. nearly. 10. If the base of a triangle be 40, and the other two sides 30 and 20; what is the length of its perpendicular? 11. Ans. 14.52 &c. If the base of a triangle be 40, and the two sides 30 and 20; what are the segments of the base made by a line bisecting the vertical angle? Ans. 24 and 16. 12. If the diameter of a circle be 30; what is the side of the inscribed equilateral triangle? Ans. 25.98 nearly. 13. If the side of an equilateral triangle be 10; what are the radii of the inscribed, and circumscribing circles? Ans. 2.8868 and 5.7736 nearly. 14. The side of a square being 10; then what is the radius of its circumscribing circle? Ans. 7071 &c. 15. If the side of a regular pentagon be 10; what are the radii of its inscribed, and circumscribing circles? Ans. 6 882 and 8.506 nearly. 16. If the radius of a circle be 10; what are the sides of the regular inscribed trigon, tetragon, pentagon, hexagon, octagon, and decagon ? Ans. 17.32-14 142-11.756-10-7-654-6.18, nearly. 17. A plan of a fortified town has a scale of 100 toises which is 1.6 inches in length; the plan is 30 inches long, and 24 broad; now what will be the size when it is copied to a scale. of 6 inches the English mile? Ans. 13'6 in. long, and 10.9 broad. 18. If the length of a pair of proportional compasses be 7 inches; how far from the ends is the centre answering to the division 5 on the line of Lines? Ans. 1 and 5 inches. 19. Suppose the length of a pair of proportional compasses to be exactly 9 inches; how far from the ends must the centres be for enlarging or diminishing a plane surface twice, and a solid three times ? Ans. 3.728 and 5.272 in. in the former case, 3.685 and 5.315 in. in the latter. 20. If the length of a cannon be 8 f. 10 in. its diameter at the breech 19 in. at the mouth 14 in. at what distance would the outer surface meet the axis of the bore supposing both were produced? Ans. 25 feet, from the muzzle. 21. How many degrees, &c. are contained in that arc of a circle whose length is equal to the radius? Ans. 57° 295779 nearly. 22. If the line of numbers from 1 to 10 on a logarithmic or Gunter's Scale is a foot; required the distance from 1 to 5.And what is the distance from 10 on the line of numbers to 40° on the line of tangents? Ans. 8.3876 &c. and 0.914 &c. inches. 23. The length of a line of chords of 90° being 44 inches; then what is the length of 45° on the same line? Ans. 2.3 in. nearly. 24. If the radius of a circle be 20; what are the lengths of the sine, cosine, tangent, cotangent, secant, and cosecant of 30° ? Ans. 10-17∙32—11·547—34·641—23·094—40. 25. If the base of a right-angled triangle be 4, and the perpendicular 3: what are the lengths of the sine, cosine, tangent, and cotangent of the least angle, if the radius be 1? 26. If the base of a right-angled triangle be 0.28, and the adjacent acute angle 59° 11'; what are the other sides? Ans. 0.5466, and 0.4694. 27. The base of a right-angled triangle being 747 yards, and its opposite angle 21° 13'; what are the other sides? Ans. 1924, and 206-4 yds. 28. The hypotenuse of a right-angled triangle being 5472 feet, and one of the acute angles 29° 51'; then what are the other sides? Ans. 4746 and 2723.5 feet. 29. If the three angles of a plane triangle are 106° 41′, 46° 24′, and 26° 55', and the side opposite the greatest angle = 297-6 yds. then what are the other sides? Ans. 225, and 140-7 yards. 30. Suppose the angles of a plane triangle to be as in the preceding example, and the side opposite the least angle 297.6 feet; required the other sides? Ans. 4761 and 629′7 feet. 31. The hypotenuse of a right-angled triangle being |