In the Appendix, the author gives a table of Annuities for three joint lives, of all combinations of quinquennial ages, from age 5 to age 100. This is also a valuable table and one very much wanted. It is exclusively confined to annuities for combinations of quinquennial ages, but it may be employed in order to find the annuities for combinations of intermediate ages, which is effected by a method of interpolation given in the introduction to this table and illustrated by some examples.

We have carefully examined these several Tables, and considering the extent to which the Anti-Logarithmic Table is carried-its convenient arrangement and beautiful typography,-it far surpasses any similar table which has come under our observation, and we have no hesitation in recommending it as a great acquisition to those whose researches involve much logarithmic calculation. The other short but useful tables are also deserving of commendation, and we have the satisfaction of knowing that the opinion we have expressed of Mr. Filipowski's Tables is in accordance with that of Professor De Morgan in his testimonial prefixed to the volume.

THE PROCESS OF THOUGHT ADAPTED TO WORDS AND LANGUAGE. BY

ALFRED SMEE, F.R.S. (London: Longman and Co.)

THIS Volume constitutes a further contribution to the electro-biological series of works which the learned author has already published. The term Electro-biology literally signifies neither more nor less than the relation of electricity to the vital functions; and the volume before us being a practical application of the subject, may possibly be immediately useful. The process of thought, and the artificial system of reasoning detailed in the work will be found of service, especially as by the right application of what the author calls the relational system, no form of sophistry or quibble can be successfully employed.

GENERAL EXAMINATION OF CANDIDATES FOR CERTICATES OF MERIT.

EASTER, 1851.

SCHOOL MANAGEMENT.

Three Hours allowed for each Paper.

SECTION. I.-1. State at length what you understand by the term " Schoolmanagement."

SECTION II.-1. In a school of 150 boys, say exactly how you would arrange five classes-a For a reading-lesson; b For writing; c For arithmetic; to be going on at the same time.

2. Show the use and abuse of the Black Board.

3. Name the different methods in which writing is taught in our Elementary Schools. State which you believe to be the best, and give your reasons. SECTION III-1. Define carefully "Notes of a lesson," state how you prepare them, and show on paper their mechanical arrangement.

2. Show, by simple instances, the difference between giving a lesson to a class ignorant of the subject, and examining the same class when in some degree informed on it.

3. In what way, and to what extent, do you instruct your apprentices in the art of teaching? Describe this carefully.

SECTION IV.-Write notes of a lesson on one of the following subjects: -Filial affection-Self-denial-Falsehood-Loyalty-Wheat-Soap-Sugar -Cotton-King Alfred-Christopher Columbus-William ShakspeareCharles I.

SECTION V.-1. Describe at length your method of giving and correcting an exercise in "dictation."

2. State how you deal with children of the following description respectively-ignorant, inattentive, rude, deceitful, unpunctual, irregular in attendance.

3. State fully and accurately the part which you take in the work of your School.

MUSIC.

SECTION I-1. In a school of 110 children, with three pupil teachers, in their fourth year, how would you teach vocal music?

SECTION II-1. Explain fully the terms Dal Segno, Da Capo, Mezzo Staccato, Fine, Legato, Sforzato, Andantino con moto, Allegretto con brio. Tutti.

2. Explain Alla Breve Time, Duple Time; and state how many beats could be made to a bar of and to a bar of Time. Write a short passage of music, to show the most convenient form for a bar of g Time.

៖

3. Are there any modulations more common than others? If there are, name them. What is a modulation?

SECTION III.-How is it known whether a piece of music is written in the major scale indicated by its signature, or in its relative minor? 2. In every major scale what notes invariably bear major thirds? 3. What is a canon? What is a Round? How are they performed? SECTION IV.-1. How many chords are derived from the common chord ? 2. What is meant by contrary motion, similar motion, oblique motion? 3. Write the chords indicated by the figuring in the following passage?

SECTION V.-Write four parts to the following passage :

MECHANICS.

SECTION 1.-1. Explain the method of estimating work done. Find the work performed by a horse which draws a ton of coal up a hill, one mile long, and rising regularly at the rate of 7 feet in every 100 yards, supposing friction to be equal to 4th of the weight of the load.

272

GENERAL EXAMINATION OF TRAINING SC

In the Appendix, the author gives a table lives, of all combinations of quinquennia' This is also a valuable table and one sively confined to annuities for ce it may be employed in order intermediate ages, which i in the introduction to th We have carefully the extent to whic'

venient arrange
any similar ta
no hesitatio

researches
useful t

satisfa Filip

in

T

composition and resolu

be proportional and parallel (a) sides, their resultant will be

its foot upon a horizontal plane, -efficient of friction on the vertical plane 7; the centre of gravity of the

vesti, determine the greatest inclination to the ver

11-bremplify the

d" oppsite to each other.

Vi spberical balls of

proposition that action and reaction are

equal size, but weighing respectively 20 and 30 second: determine their movements after collision, sup

es, more directly towards each other with the respective velocities of 75

and 17 feet in a

the force of impact.

acquired, and the space described in a given time.

city of 1000 feet in a

From the top of a tower, 170 feet high, a shot is fired with a vertical velo2. A body is moved from rest by the force of gravity; find the velocity relocity with which it would fall to the ground, if the air offered no resistper minute, the fall of the water 14 feet; to what height will it raise a cubic If the modulus of one of these machines be '87, the water spent 18 gallons 3. Describe the construction and action of the hydraulic ram. foot of water in 80 seconds.

second; find the height to which it would rise, and the

ance to its passage.

EUCLID.

SECTION 1.-1. If two angles of a triangle be equal to each other, the sides also which subtend the equal angles shall be equal to each other.

If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to each other: and the exterior angle equal to the interior and opposite upon the same side and likewise the two interior angles upon the same side together equal to two right angles.

3. To describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle.

SECTION II.-1. If a straight line be divided into two equal parts, and also into two unequal parts: the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.

2. To describe a square that shall be equal to a given rectilineal figure. 3. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

SECTION III-1. In a circle the angle in a semicircle is a right angle. 2. If two straight lines cut one another within a circle, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other.

3. To describe a square about a given circle.

SECTION IV.-1. Triangles and parallelograms of the same altitude are to each other as their bases.

2. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to each other.

3. If two opposite sides of a parallelogram be bisected, and two lines be drawn from the points of bisection to the opposite angles, these two lines trisect the diagonal.

SECTION V.-1. Describe an isosceles triangle, each of the sides of which shall be double of the base.

2. Show that in a parallelogram the sum of the squares of the diagonals is equal to the sum of the squares of all the sides.

3. The arcs intercepted between any two parallel chords in a circle are qual.

HIGHER MATHEMATICS.

SECTION I.-1. The sum of an arithmetic series is 440, the first term 3, and the common difference 2. What is the number of terms?

2. Prove the rule for the number of permutations of n different things taken all together. In how many different ways may the letters forming the word Buckingham be arranged?

3. Prove that if the sum of the digits of any number be divisible by 3, the number itself is divisible by 3.

SECTION II.-1. Investigate a rule for finding the present value of £ a, due n years hence, r being the rate of interest.

2. A person is entitled to an annuity of £100, commencing three years hence, and continuing for three years. He wishes to exchange it for an annuity to commence immediately, and terminate at the same time with the other annuity. The rate of interest being 5 per cent.; to what amount of annuity is he entitled ?

3. What is a logarithm? Find a logarithmic expression for the value of x in the equation

a bx + c
d

= e.

SECTION III.-1. Define the secant of an angle, and find its value in terms of the sine.

3. Given an angle (A) of a triangle, and the sides (b, c) containing it, investigate formulæ for the values of the opposite side (a), and the remaining angles (B, C).

SECTION IV.-1. Prove that the sides of a triangle are proportional to the sines of the opposite angles.

2. Find the value of sin. 18°.

3. The elevation of a tower is observed from a certain distance. At a station a feet nearer, the elevation is the complement of the former; b feet nearer still, it is double the first elevation. What is the height of the tower? SECTION V.-1. Explain what you mean by a system of coordinates, and by the equation to a line, and find the equation to a straight line which cuts the axis of x at a distance from the origin = a, and the axis of y at a disb, the system of coordinates being rectangular.

tance

2. What do you mean by a conic section? Find the equation to the ellipse, the extremity of the axis major being the origin of coordinates, and the axis major being on the axis of x: and from it deduce the equation when the origin is at the centre of the ellipse.

3. What do you mean by maxima and minima of a function of a variable. Investigate a rule for their discovery, and find the largest possible area of a right-angled triangle, the hypothenuse of which c..

2. Explain, as you would to a Pupil Teacher, the composition and resolution of Forces.

Show, that if the forces, acting upon a body, be proportional and parallel to (n-1) consecutive sides of a polygon of (n) sides, their resultant will be represented by the nth side.

3. A ladder, 30 feet long, is placed with its foot upon a horizontal plane, and its top against a vertical plane; the co-efficient of friction on the vertical plane is 3, that on the horizontal plane 7; the centre of gravity of the ladder is 12 feet from its foot; determine the greatest inclination to the vertical at which it will rest.

SECTION II.-Exemplify the proposition that action and reaction are equal and opposite to each other.

Two spherical balls of equal size, but weighing respectively 20 and 30 ounces, move directly towards each other with the respective velocities of 75 and 17 feet in a second: determine their movements after collision, supposing the force of elasticity to be the force of impact.

2. A body is moved from rest by the force of gravity; find the velocity acquired, and the space described in a given time.

From the top of a tower, 170 feet high, a shot is fired with a vertical velocity of 1000 feet in a second; find the height to which it would rise, and the velocity with which it would fall to the ground, if the air offered no resistance to its passage.

3. Describe the construction and action of the hydraulic ram.

If the modulus of one of these machines be '87, the water spent 18 gallons per minute, the fall of the water 14 feet; to what height will it raise a cubic foot of water in 80 seconds.

EUCLID.

SECTION 1.-1. If two angles of a triangle be equal to each other, the sides also which subtend the equal angles shall be equal to each other.

2. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to each other and the exterior angle equal to the interior and opposite upon the same side and likewise the two interior angles upon the same side together equal to two right angles.

3. To describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle.

SECTION II.—1. If straight line be divided into two equal parts, and also into two unequal parts: the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.

2. To describe a square that shall be equal to a given rectilineal figure. 3. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

SECTION III.1. In a circle the angle in a semicircle is a right angle. 2. If two straight lines cut one another within a circle, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other.

3. To describe a square about a given circle.

SECTION IV.-1. Triangles and parallelograms of the same altitude are to each other as their bases.

2. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to each other.

3. If two opposite sides of a parallelogram be bisected, and two lines be drawn from the points of bisection to the opposite angles, these two lines trisect the diagonal.