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of the flange is 3 feet; the face of the wheels to be so bevelled that at the outside of each wheel the diameter of the wheel is 2 feet 11.5 inches, and that the axle will always be in a position square across the two rails. In what part, between the two wheels, must the centre of gravity of the load be placed, so that the weight of the load shall bear equally on each rail ?

OPERATION. 5ft. 4in. - 5ft. = 4in.; and 4in. + 2 = 2in. = the place, be. yond the flange, on the face of the wheel, where the wheel bears on the rail. The perpendicular width of the face of the wheel is 4 inches, and in that distance the wheel diminishes in diam eter 3ft. - 2ft. 11.5in. = .5in. Then, as 4in. : .5in. : : 2in. : .25in.; and so the diameter of the wheel at the bearing is 3ft. —.25in. = 2ft. 11.75in. As the radius of the curve of the outer rail is to that of the inner rail, so is the diameter of the outer wheel at the place of bearing to that of the inner wheel; viz. as 1200ft. : 1200ft. - 5ft. 4in. = 1194ft. 8in. :: 2ft. 11.75in. : 2ft. 11.59lin. Then the difference of the diameters of the two wheels at the places of bearing is 35.75in. — 35.59i= .158in. Then, to see how much on the perpendicular face of the wheel this difference in diameter will vary the bearing; as 3ft. - 2ft. 11.5in. =.5in. : 4in. : : .158in. : 1.27lin. But this difference must be applied half to each wheel, viz. 1.271in. + 2=.635in., which brings the place of bearing .635in. nearer to the flange on the outer wheel, and .635in. further from the flange on the inner wheel. And the middle point between the two bearings will be .635in. from the centre of the axle towards the inner curve; and in that place must the centre of gravity of the whole load be placed, to make the weight of the load bear equally on each rail ; viz. .635in. from the middle of the axle towards the inner rail, Answer.

36. The dimensions of a bushel measure are 184 inches wide, and 8 inches deep; what should be the dimensions of a similar measure that would contain 8 bushels ?

Ans. 37in. wide, 16in. deep. 37. What is the weight of a hollow spherical iron shell 5in. in diameter, the thickness of the metal being lin., and a cubic inch of iron weighing 28 of a pound ? Ans. 13.23871bs.

38. At a certain time between 2 and 3 o'clock, the minute hand was between 3 and 4. Within an hour after, the hour hand and minute hand had exactly changed places with each other. What was the precise time when the hands were in the first position ?

Ans. 2h. 15m. 569 sec. 39. Required the contents of a cube, that will contain a globe 20 inches in diameter; also of a cube, that may be inscribed in said globe.

Ans. Larger cube 8000 cub. in.; smaller, 1539 cub. in. 40. If in a pair of scales a body weigh 90 pounds in one scale, and only 40 pounds in the other, required the true weight, and the proportion of the lengths of the two arms of the balance. beam on each side of the point of suspension.

Ans. the weight 60lbs., and the proportions 3 to 2. 41. In turning a one-horse chaise within a ring of a certain diameter, it was observed that the outer wheel made two turns, while the inner wheel made but one; the wheels were each 4 feet high; and supposing them fixed at the distance of 5 feet asunder on the axletree, what was the circumference of the track described by the outer wheel ? Ans. 62.83+ feet. · 42. The ball on the top of St. Paul's church is 6 feet in diameter. What did the gilding of it cost, at 3.1d. per square inch?

Ans. 237 £. 10s. ld. 43. There is a conical glass, 6 inches high, 5 inches wide at the top, and which is part filled with water. What must be the diameter of a ball, let fall into the water, that shall be immersed by it ?

Ans. 2.445+in. 44. A certain lady, the mother of three daughters, had a farm of 500 acres, in a circular form, with her dwelling-house in the centre. Being desirous of having her daughters settled near her, she gave to them three equal parcels, as large as could be made in three equal circles, included within the periphery of her farm, one to each, with a dwelling-house in the centre of each; that is, there were to be three equal circles, as large as could be made within a circle that contained 500 acres. How many acres did the farm of each daughter contain, how many acres did the mother retain, how far apart were the dwelling-houses of the daughters, and how far was the dwelling-house of each daughter from that of the mother?

Ans. Each daughter's farm contained 107 acres 2 roods 31.22+ rods. The mother retained 176 acres 3 roods 26.34+ rods. The distance from one daughter's house to the other was 148.119817+ rods. The mother's dwelling-house was distant from her daughters' 85.51+ rods.

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47. Multiply the cube root of i by the square of 1

Ans. 125 4 145 655, 134 2311 48. Add 213 of 98 to 311 of 37*

Ans. 731. 49. Three carpenters, J. Smith, J. Carleton, and John Jones, agree with T. Jenkins to build him a house and find the materials for $ 1000, of which $ 600 were paid in advance, and the remainder when the work was finished. Carleton and Jones took $ 50 each of the first payment. When the work was completed, it appeared by Smith's account, who received the money and paid the bills, for which he was allowed a compensation of $ 10, that he had paid $ 648.95, exclusive of the payments to Carleton and Jones, and that he had labored 63 days. Carleton worked 51 days, and he was allowed $ 20 for the use of his shop, &c. Jones worked 60 days, and his bill for boarding the men they hired was $ 68.75. Smith, on settling with Jenkins and allowing him $ 23.15 charged to Carleton, and $ 17.48 charged to Jones, receives the balance in cash, and on exhibiting his statement of the business to Carleton and Jones, he pays to each the balance due. How much did they make per day, and how was the last payment disposed of?

Ans. They received $ 1.45 per day, and Carleton received $ 20.80, Jones received $ 88.27, and Smith received $ 250.30.

50. A and B engaged to reap a field for 90 shillings; and as A could reap it in 9 days, they promised to complete it in 5 days. They found, however, that they were obliged to call in C, an inferior workman, to assist them for the last two days, in consequence of which В received 3s. 9d. less than he otherwise would have done. In what time could B and C each reap the field ? Ans. B could reap it in 15 days, and C in 18 days.

51. Samuel Jenkins and James Betton, who have each an apprentice, engage to build a small house for $ 630. By agreement between them, Jenkins's apprentice is to be allowed $0.624 per day, and Betton's $ 1.00. When the work was finished, it appeared that Jenkins had worked 120 days, and his apprentice 100. Betton worked 96 days, and his apprentice 135,1 days. While doing the work, they received each $ 210 What is each man's share of the remaining payment ?

Ans. Due to Jenkins $ 92 50 ; to Betton $ 117.50. 52. A merchant tailor bought 40 yards of broadcloth, 21 yards wide ; but on sponging it, it shrunk in length upon every 4 yards half a quarter, and in width, one nail and a half upon every l} yards. To line this cloth, he bought flannel 5 quarters wide, which, being wet, shrunk the whole width on every 20 yards in length, and in width it shrunk half a nail. Required the number of yards of flannel used in lining the cloth.

Ans. 7115 yards. 53. What is the square of .i ?

54. If a stick of timber, 6 inches square and 10 feet long, will support from its centre 1000 pounds, how many pounds would a similar stick, that is 10 inches square and 20 feet long, support, if the weight were suspended 4 feet from the centre of the stick?

55. A certain gentleman has an elegant lamp, to which are appended 72 brilliants, each of which had occupied a particular place. His daughter, one day, in preparing the lamp, dis. placed the brilliants, and in replacing them, found that she had put some of them in the wrong situation, and was at a loss to know how to correct the mistake. At length she told her father that, after dinner, she would begin and place the brilliants in all the situations they would admit of, and then she would be sure of finding the correct way of adjusting them, and that she would not take her tea until she had effected it. Now, supposing she were to place them in all the various ways they would admit, how long would she be obliged to wait for her tea, provided she could make one change each minute ?

Ans. 612344583768860868615240703852746727407780917 8469732898382301496397838498722168927420416000000000 0000000 minutes.

56. I have a garden in the form of an equilateral triangle, whose sides are 200 feet. At each corner stands a tower; the height of the first is 30 feet, the second 40, and the third 50. At what distance from the base of each tower must a ladder be placed, that it may just reach the top of each ? and what is the length of the ladder, the garden being a horizontal plane ?

Ans. The foot of the ladder from the base of the first tower 118.8117 feet; second tower, 115.827+ feet; third tower, 111.875+ feet. Length of the ladder, 122.535+ feet.* * The operation of this example may be found on page 153 of the Key.

APPENDIX.

WEIGHTS AND MEASURES,

WITH AN HISTORICAL ACCOUNT OF STANDARDS

In commerce and in science, all bodies are estimated by number, weight, or measure. When reckoned by number, they are referred to a unit as a standard of comparison, and when estimated by weight or measure, there is always a reference to some certain fixed quantity, as a pound, a gallon, a mile, to which the quantity measured or weighed bears a specified and definite proportion.

Weights are used to ascertain the gravity of bodics, and measures to determine their magnitude, or the space which they occupy.

Standards of weights or measures are certain quantities of gravity or extension, which are fixed upon as those with which all other quantities of objects reckoned by weight or measure are to be compared; and such standards have always been found necessary, and have existed in every age and nation. It has, however, been only in a highly civilized state of society that they have been such as to secure an accurate and equitable result in the transaction of business; and few things are more indicative of a cultivated age and people, than the exactness with which science provides and adjusts the standards of comparison, required by the sale and interchange of commodities.

In the early stages of society, the ordinary standards of weight and measure were some simple objects or ideas, with which all in the community were supposed to be familiar. Thus, all measures of length were sometimes reckoned by comparing them with the human foot; or, for the sake of greater definiteness, as all human feet were not of the same length, they were referred to the king's foot as a standard. In some cases the . length of the arm was used, and in other instances the length of a grain or corn of wheat or barley. In land measure, an acre was what could be ploughed by a yoke of oxen in a day, traces of which notion we have in the Hebrew and Latin words used to denote an acre, which properly signify a yoke or pair, - that is, a yoke or pair of oxen.

In early ages, also, weights as well as distances were meas

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