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3. May 25th, 1850, in latitude 9° 35' S. nearly, longitude 144o E., the following observations were taken near noon to determine the latitude :

h. m. 8. 11 58 34

118 29 40
11 59 14

30 20
App. time
11 59 55

30 Index error + 30".
12 046

29 40 12 1 37

29 20 Required the latitude.

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SECTION V:-1. Apparent distances of centres of O and ) 107 0 19

Apparent altitude of o’s centre 25 52
True

25 50 12 Apparent altitude of j's centre 8 58 0 True

9 45 51 Compute the true distance of the centres,

2, December 29th, 1850, the following observations of equal altitudes were made at Funchal, Latitude 32° 38' N. Longitude 16° 55' W.

h. m. s. A.M.

10 51 438 P.M.

2 57 16:1 Determine the error of the chronometer from mean time at place; and also the longitude; the chronometer on the 18th of December being slow of mean time at Greenwich 13m 8*, losing daily 0:34*.

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Lower School.- 1st Class.
Section I.—Prove the following rules:

rad : sec. lat. :: any portion of parallel : like portion of equator.
Cos, mid. lat. : dist. :: Sin. course : diff. long.

diff. lat. : diff. long. :: Cos. mid. lat. : tan. course. Section II.-1. Describe and explain the sextant, and show how its error may be determined.

2. What is meant by sidereal time, and what by mean solar time; from what cause does the difference between them arise, and how may one be calculated from the other ?

3. Investigate the formula for determining the latitude from an observed altitude of the sun; the mean time at the place and the longitude being given.

SECTION III.-1. A port bears S, 52° W., the current sets S.S.E. 2 miles an hour; the present rate of sailing 7 knots : shape the course so as to keep the port on the same bearing.

2. Find the course and dist. between C. Sierra Leone in lat. 8° 30' N., long. 13° 8' W. and C. San Roque, lat. 5° 28' S., long. 35° 17' W., 1st by mid. lat. sailing ; 2nd, by Mercator's sailing ; 3rd, by great circle sailing.

SECTION IV.-1. On the 24th of May, 1851, in long. 25° W. the meridian altitude of the sun's lower limb was 810 20' 3" towards the south. Height of eye 33 ft., index error + 1' 30". Find the latitude.

2. October 15th, 1850, at 85, P.M. on the coast of Spain, and nearly on the meridian of Greenwich, observed ) 39° 49' 30"; index error + 35" ; height of eye 20 feet. Required the latitude of the place; the supposed latitude being 34° north.

3. July 21st, 1850, the setting amplitude of O in latitude 27° 32' S., longitude 170° E., was W. 23° 15' N. by azimuth compass. Required the variation. (Greenwich date 20d. 18h.)

SECTION V.-1. Sept. 28th, 1850. Of the Ferroe Islands, lat. 68° 7' north, at 4b P.M.

Time chron. 3h 53m 49.89

0 10° 8' 12" Index error -- 4 50". Height of eye, 17 feet. Determine the errors of the chronometer from Greenwich mean time ; assuming the longitude of place of observation to be 70 55' 0" west. 2. O's true alt. 36° 16' 14"

D's true alt. 12° 12' 10"
O's app:. alt. 36° 17' 24"

D's appt. alt. 11° 22' 13' apparent distance of centres of O and ), 66° 27' 46", required the true distalice. 3. On 5th November, 1851, { Long. 14° E.

Lat. 35° 53 50" N. } at 847 26' 4.M. apparent time, compute the apparent altitude of the sun's centre.

Lower School.-1st and 2nd Classes. Section 1.-1. If at a point in a 'given straight line, two other straight lines on the opposite side of it make the adjacent angles together equal to two right angles, these straight lines are in one and the same straight line.

2. To describe a square which shall be equal to a given rectilineal figure.

3. Upon a given straight line to describe a segment of a circle which shall contain an angle equal to a given rectilineal angle. SECTION II.-Solve one of the following equations :2x + 3 4x

1 6x + 2
1.
3

3

x + 1

+

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4

+ X

6

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X

3

x + 3y 31

3 1 4.

+ XC + 2 2 20 1° Section III.-1. A draper bought some pieces of linen for 63l., and having reserved two, disposed of the remainder for 60l., clearing 10s. on every piece he sold. How many pieces did he buy?

2. I have a number of halfpence which I try to arrange in a square; on the first trial I find that I have 130 over; I then enlarge the side of the square by 3 and find that I have 31 over. How many halfpence have I?

3. Two men have to plant a trees in a straight line b feet apart, carrying each tree separately from a heap at one end of the line. How shall they divide the line so that one planting one part of it and the other the othe part, they may have to walk equal distances ? Section IV.-1. Prove the following formulæ of trigonometry :

rada

rad? rad? + tan coseca

Cot

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cec =

COS

tan

2. Show that in any right-angled triangle ACB, whose right angle i C, BC

AC tan A. 3. Cos (A + B) Cos A Cos B Sin A Sin B.

SECTION V.-1. A labourer working with a wheel and axle 8 hours a day can yield at the rate of 2600 units of work per minute. How much must be charge per ton for raising coals from a depih of 25 fathoms that he may earn 2s. 6d. per day?

2. The section of a stream is 5 feet by 7, and its mean velocity 15 feet per minute ; there is a fall upon it of 11 feet, working a wheel which yields •6 of the work the water is capable of doing ; how many quarters of corn is the mill capable of grinding, allowing a bushel per hour per horse power?

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3. How many horses, each exerting a traction of 200 lbs., would be required to draw a waggon, weighing, gross, 5 tons, up a hill whose inclination is 1 in 18; the traction on the level road being estimated at 1-20th of the load ?

Section VI.-1. How is it known that the earth is not an infinitely extended plain, as it seems to be? Give one reason, and that the simplest.

1. Explain the phenomena of the seasons.

3. What principal change would appear in the heavens to a person who watched them at the same place night after night for a year ?

A sentence will be read to you, which you are to write down carefully, and spell correctly.

Nautical School. Section I.—Correct the following sentences, and assign the grammatical reasons for the corrections you make :

“ His friends has forsaken him." No man is more happy than him.” “ It is me.” “We ought always to act as justice and honour requires." “The fleet is all arrived in safety.” Thou which has been a witness of the fact, can give an account of it.”

I hope it is not I who you condemn.” SECTION II. Describe the most remarkable periodical winds, and account for them. What law has been observed in hurricanes ?

2. What are the prevalent currents of the ocean? Account for them. 3. What are the principal food of plants ? Give some account of their distribution on the earth's surface.

SECTION III.-1. Write down what you know of France.

2. What countries compose the British Einpired By what races of men are they inhabited ? How are they governed ? In what does the commerce of each chiefly consist ?

3. Give some account of the political divisions of South America, and of its physical geography.

SECTION IV.-1. How many yards of cloth, at ll. 35. 6d. per yard, can I buy for 501. ?

2. What is the difference between šof a cwt. and s of a ton?

3. Two clocks point out 12 at the same instant: one of them gains 7" and the other loses 8" in twelve hours: after what time will one have gained half an hour on the other, and what o'clock will each then show ?

4. How many cubical feet of planking, & inch thick, will it take to make the floor of a room whose dimensions are 36 feet 7 inches by 22 feet 11 inches ?

SECTION V.-1. The two sides AC and AB of a triangle are respectively 514 and 317 feet in length, and the included angle A is 19° 15' 3". Find the side BC. 2. Find the value of the following expression by logarithms :

♡ 5:1362 x

N.0275 2:3527

а

Lower School.3rd Class. A sentence will be read to you, which you will write down carefully and spell correctly.

1. Draw a map of England.

2. Write down in words the number represented by the following figures : -30,050,601.

3. How many years was it from the building of Solomon's Temple, in 1012 B.C., to that of St. Paul's Cathedral, in 1688 A.D. ?

4. Multiply 357,962 by 2705.

5. Subtract 359l. 11s. 64d. from 4001.

6. The difference of two numbers is 171,321, and the greater is 479,328 ; find their product.

7. Having a sum of 5000l. to lay out, I buy with it three houses, one of which costs me 10297. 15s. 6d., another 1165l. 14s. 8d., and the third 25637. 11s. 9d. How much have I left?

8. In 375,698 lbs. how many tons ?

9. If 68 cwt. 1 qr. 18 lbs. cost 286l. 14s. 11 d., what will 3 lbs. cost? 10. A coach wheel which is 9 ft. 6 in. in circumference, makes 14 revolutions in 15 seconds; at what rate per hour is the coach travelling?

Lower School.-4th Class.

A sentence will be read, which you are required to write down carefully and spell correctly.

1. Correct the grammatical construction of the following sentences :— Those set of books was a valuable present."

66

"This is he who I gave the book to."
"William and I am cousins."

2. Draw a map of England.

3. Multiply 92948789 by 7043.

4. Divide 75843639426 by 8593.

5. Reduce 3717. 14s. 10d. to halfpence.

6. Find the 11th part of 71. 8s. 64d., and the 9th part of 361. 4s. 9d. 7. How much money must I add to 3401. 7s. 9d. to make the sum of 500l. ?

And how much must I take from 59l. 13s. 6d. to make the remainder 187. 11s. 5d.?

8. What is the price of 72 reams of paper at 13s. 8d. per ream?

9. How many dozen bottles of wine may be filled from 15 pipes, each pipe containing 53 dozen 7 bottles?

10. If 47 lbs. cost 347. 10s. 3 d. what will 1lb. cost?

11. If 2 cwt. 46 lb. cost 137. 10s., how many cwt. may be bought for 57. 12s.?

12. Add together

and.

ANSWERS TO THE MATHEMATICAL QUESTIONS.

QUES. 107. Proposed by Sam. Dyer,

81

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33 to find a.

Ꮖ 33x

Answered by Mr. Hill, Mark, Mr. Morris, Mr. Abbott, Mr. Henry, and the Proposer.

By multiplication and transposition.
36x 82 + 33x

completing the square.

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36x 82 × 33 + 412 ... 33x-41 ** ვ3a Hence we have the two equations. 33x = From the first equation, we get

=-81,

81 + 412
± 40,
81 or 1.

=

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34, and 33x

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3x = 4, .. x = x = 1/3, From the second, 3 = 0, or x = 0.

Ques, 108.-- Proposed by Prismoid, Tewkesbury. Six equal circles are described within a given circle, in contact with one another and the given circle ; required the area of each.

Answered by Mr. J. Sheppard, Mr. Sothern, Mr. Righton, II.P.,

A.M., und the Proposer. The six equal circles will be inscribed in the six equilateral triangles formed by joining the centre of the given circle, and the ungular points of a hexagon circumscribed about. Letr = the radius of the given circle; then

The height of each of the trianglis = '',

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Ques. 109.- Proposed by Mr. Herbert. At what time between 11 and 12 o'clock, are the hands of a clock approaching the 12 with the same velocity?

Answered by T. T. T., the Proposer, and Mr. Sheppard. Let 2 x the No. of degrees, on the dial, that the short hond is from XII ; then 24 x the No. of degrees that the long one is from XII. The rectilineal distance of the short hand = 2 Sin x, and

long hand = 2 Sin 12.x. Now the differential coefficient of these distances will give ratio of the velocities of the two hands in their approach to the 12 upon the dial; bence we have

d. (2 sin x)

= 2 cos x,
dc
d. (2 sin 12x)

24 cos 12x,

dx

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which gives the equation from which x may be determined ; but as the general solu. tion of this equation involves considerable difficulties, we shall determine x by the following easy method of approximation.

As the short pointer must be less than five minutes from the XII, it follows that x must be less than 90 from this point. Now the cosine of 9° is very nearly unity; hence we get very nearly

12 cos 12 x = 1,

1
... cos 12 x = •083333,

12
.. 12 x

85° 13', and 24 x = 170° 26' nearly, which is the distance, in degrees, of the long pointer from XII. Reducing this angular distance to minutes of time, we get the required time to be 287% min. before XII.

Ques, 110.- Proposed by Mr. Righton. Determine the segment of a sphere so that the ratio of its solidity to its convex surface may be a maximum.

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