ANSWERS TO THE MATHEMATICAL QUESTIONS.

QUES. 135.-Proposed by Prismoid, Tewkesbury.

I retired to rest one evening at m minutes to h o'clock, and on rising 6 hours and 55 minutes after, it wanted h minutes to m o'clock. Required the time of rising.

Answered by Mr. Sothern, and similarly by Mr. Levy, A.M., Mr.

=

Salter, and Mr. Abbott.

... h -m= 5. An indeterminate equation.

Hence h may 6, 7, 8, 9, 10, 11, or 12; and m may = 1, 2, 3, 4, 5, 6, or 7. He may therefore go to rest at any of the following times; viz., 1 min. to 6, 2 min. to 7, 3 min. to 8, and so on; and he may rise at the following corresponding times, viz., 6 min. to 1, 7 min. to 2, 8 min. to 3, and so on.

QUES. 136.-Proposed by Mr. Collins, Liverpool.

A man has a calf, which, at the end of four years, has its first calf; at the end of three years after that, has its second calf; and at the end of every two years after that, has a calf. Each calf also, like the original one, when four years old has its first calf; at the end of three years after that, has its second calf; and at the end of every two years after that, has a calf. How many head of cattle will the man have at the end of 30 years?

N.B.-Not having received a correct_answer to this question, its solution is deferred until next month.

QUES. 137.-Proposed by Mr. O. Clazey, Shincliffe.

A tree, in the form of a frustrum of a cone, is 15 feet long, and the diameters of its greater and less ends are five and three feet respectively. Required the length of the greatest square beam which can be cut out of the tree.

Answered by the Proposer, and similarly by Mr. Dyer, Mr. Salter, and Mr. Righton.

Let y be the length of the required beam, and 2 x the side of its end; then by similar triangles,

A.M., Cambridge, ans. 135, 137; J. Salter, Durham, ans. 137; T. Sothern, Burtonwood, ans. 135, 137; W. Righton, ans. 137; E. Rutter, ans. 137; T. W. Haigh,

Huddersfield, ans. 135; S. Dyer, Wanstead, ans. 135, 137; O. Clazey, Shincliffe, ans. 135, 137; W. H. Levy, Shelbourne, ans. 135, 137; W. Abbott, Twickenham, ans. 135; J. Rowlatt, Evercreech, ans. 135, 137; T. Horsman, Chelsea, ans. 137.

NEW QUESTIONS,

TO BE ANSWERED IN OUR NUMBER FOR AUGUST, 1852.

QUES. 138.-Proposed by Mr. Abbott, Twickenham.

In how many days of 8 hours long will a man with a cart and horse transport a cubic feet of earth to the mean distance of m miles, supposing the horse to travel with the full load of b cubic feet, at the rate of per hour, and to return with the empty cart at the rate of hour?

QUES. 139.-Proposed by Mr. Levy, Shelbourne.

miles

miles per

A cylindrical glass, six inches long, is forced into water with its mouth downwards, until the water rises one inch in the glass; required the depth to which the glass is depressed in the water.

QUES. 140.-Proposed by Mr. O. Clazey, Shincliffe.

If in a circle whose radius is 6, a right angled triangle be inscribed, show that, when a maximum circle is inscribed in the triangle, the area of the triangle is 36.

GENERAL EXAMINATION OF CANDIDATES FOR

CERTIFICATES OF MERIT.

EASTER, 1852.

EUCLID.

SECTION 1.-1. The angles at the base of an isosceles triangle are equal to each other.

2. All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

3. To a given straight line to apply a parallelogram which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

SECTION II.-1. If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part.

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

3. If A B, the hypothenuse of a right angled triangle A B C is trisected in the points D and E, prove that C D + DE6 + E C = A B2.

SECTION III.-1. The angles in the same segment of a circle are equal to one another.

2. Upon a given straight line to describe a segment of a circle, which shall contain an angle equal to a given rectilineal angle.

3. If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it, the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it.

SECTION IV.-1. To inscribe an equilateral and equiangular hexagon in a given circle.

2. If the sides of two triangles, about each of their angles, be proportionals, the triangles shall be equiangular; and the equal angles shall be those which are opposite to the homologous sides.

3. In right angled triangles, the rectilineal figure described upon the side opposite to the right angle, is equal to the similar and similarly described figures upon the sides containing the right angle.

SECTION V.-1. If two lines be drawn bisecting each other, the lines which join their extremities will form a parallelogram.

2. Construct a parallelogram equal to a given parallelogram, and having all its sides equal.

3. If two circles, A B C and A B D, intersect in A and B, and A C, AD be two diameters, prove that the line C D will pass through B.

SECTION IV.-1. A man left £560 between his son and daughter, so that, for every half crown the son should have, the daughter was to have a shilling. How much did each receive?

2. A person being asked what o'clock it was, answered that it was between six and seven, and that the hour and minute hands were exactly upon each other. What was the time of day?

3. A banker borrows money at 3 per cent. per annum, and pays the interest at the end of the year: he lends it out at the rate of 5 per cent. per annum, but receives the interest quarterly, and by this means gains £200 a-year. How much does he borrow? SECTION V.-1. A grazier buys a certain number of oxen for £80: and finds that if he had bought four fewer for the same money they would have cost £1 a-piece more. How many oxen did he buy?

2. Find two numbers such that their product shall equal the difference of their squares and the sum of their squares shall equal the difference of their cubes.

3. A person spends in the first year m times the interest of his property; in the second, 2 m times that of the remainder; in the third, 3 m times that of what is left at the end of the second year; and so on, at the end of 2 p years he has nothing remaining. Show that in the pth year he spends as much as he has left at the end of that year.

HIGHER MATHEMATICS.

SECTION I.-1. Find the sum of 20 terms of the series, 1, 1, &c.

2. Insert 10 Arithmetic means between 10 and 1000.

3. The difference of the means of four numbers in geometric progression is 2; and the difference of the extremes is 7: required the numbers.

SECTION II.-1. In how many different ways may £100 be paid in crowns and guineas?

2. Show that the number of permutations of m things taken n together is equal to m. (m − 1)... (m—r + 1) times the number of permutations of mr things taken nr together.

3. Find the sum of the series

SECTION III.--1. Calculate an Algebraical Expression for the Discount due on £A at the end of N months, supposing the rate of interest to be £R for every £100, and apply it to the case of £275 68. 8d., due 18 months hence, at 4 per cent.

2. Show, Algebraically, that a number is divisible by 9, if the sum of its digits is divisible by 9 and by 11, if the sum of the 1st, 3rd, 5th, &c. digits is equal to the sum of the 2nd, 4th, 6th, &c. digits.

3. Prove the Binominal Theorem when the index is a whole number, and show after how many terms the series for (1 + 3)10 converges.

SECTION IV.-1. Explain the construction of Logarithmic Tables, and show that, with their aid, the operations of involution and evolution may be performed by those of multiplication and division.

2. Prove that Sec A - = 1 + Tan A × Tan A, and adapt this expression to the Circle whose radius is R.

SECTION V.-1. Find the equation to a straight line, which shall pass through a given point, and be perpendicular to a given straight line.

2. The sum of the squares of any two conjugate diameters of an ellipse is a constant quantity.

3. Trace the curve, whose equation is y = sec. x, and find its area.

MECHANICS.

SECTION I.-1. Why is perpetual motion mechanically impossible?

2. Explain clearly the difference between the force of pressure and the force of a blow. 3. Explain the action of the Screw, and show the mechanical advantage gained by it. SECTION II.-A Well is 3 feet in diameter, and 60 feet deep, and full of water; what is the H. P. of an engine which empties it in 15 minutes?

2. Find an expression for the range of a projectile, neglecting the resistance of the air. 3. Explain pattern weaving.

PHYSICS.

SECTION 1.-1. Why is a stone or brick floor colder to the feet than a boarded floor? 2. What is meant by radiation and reflection of Heat respectively? Show how one who understands this may economise fuel.

3. "Can the art of Agriculture be based upon anything but the restitution of a disturbed equilibrium?" (Liebig's Letters on Chemistry).-Answer this question, and illustrate your answer by reference to the principal manures used by Farmers.

SECTION II.-1. The specific gravities of two fluids being 78 and '66, what weight of the latter fluid will fill a vessel which holds 91lb. of the former?

2. The focal lengths of the object and eye glasses of a refracting telescope being given, find its magnifying power.

3. Describe and explain the Lactometer.

SECTION III.-1. Give some general rule for the position of a black board with respect to the light, in order that the class may see clearly what is chalked upon it. 2. Explain the variety of tints which are seen at sunset.

3. Show that waves are in fact a vibration; and illustrate this from the theory of sound.

SECTION IV.-1. What are the principal means of exciting electricity?

2. What was Oersted's great discovery? Explain its practical application in the Electric Telegraph.

3. What grounds have we for believing that Heat, Light, and Electricity may be ultimately referred to one common principle?

MUSIC.

SECTION I.-1. Describe the Diatonic Scale; and write out the Notes of which it is composed in the Key of E (Mi) Natural, or in that of E Flat. Put the proper mark to each Note, according as it is sharp, Flat, or Natural.

2. Explain the Term "Clef;" and describe the different Clefs used in writing Music. Write out one Octave of the Scale of E Flat in the "Do or C Clef," as used for a Tenor part. Show the places of the same Notes if written in the "Sol" (G), and “Fa" ' (F), Clefs.

3. Put Chords to this Figured Bass; and state, if you can, to what species of composition the Passage belongs.

SECTION II.-1. Explain the Terms,-" Adagio," "Andante," " Allegretto agitato." From what language are they derived? Name any other terms connected with these. 2. Explain the Terms,-" Major Third," "Fourth," "Imperfect Fifth,' "'" Minor Sixth ;" and write out an example of each; using no sounds but such as are found in the Key of E Natural, or of D (Re), Flat.

3. Write out, in order, the Names of the Chords and Discords in this Passage; or, if you prefer it, represent them by a Figured Bass. Do you perceive any violation of the laws of Harmony?

SECTION III.-1. How may the Key, in which any Piece of Music is written, be determined with certainty?

2. On which Notes of any Scale can Major Chords be written or sounded, without the use of Accidentals? Show this, in the Keys of B (Si) Flat, and of G (Sol).

3. How many Chords are derived from the Common Chord? Give examples, with B Flat, or G, as the Key-note and fundamental Bass.

SECTION IV.-1. Give full "Notes of a Lesson " on the following Passage, under these several "Heads: "-Key-Time-Intervals-Expression. If you recognize the air, say so; and mention any changes here made in writing it.

2. If you had to instruct Pupil Teachers how to adapt a Second to this Melody, what Rules should you give them?

3. Show, as you would to Pupil Teachers tolerably advanced in the study of Harmony, how this Melody may be harmonized in three parts.

Andantino.

SECTION V.-1. In teaching young children to sing by ear, what are the faults which it is most necessary to guard against?

2. Which do you find more difficult, -to teach children to sing correctly in respect of intonation, or correctly in respect of time? Give reasons in support of your answer. 3. By what means would you try to train children to join in Congregational Psalmody; (1) In a neighbourhood where there was little taste for Music; (2) Where the prevailing style of singing was loud and boisterous?