and F, E, 4 feet, .. F, E, A, D, will be expressed by area of F, E, A, D.

3. Since each of the spaces from G to H is 1 inch (or th of a foot) in width, if they were 12 inches long, each would be a prime; but, each from H to B is only 1 inch (or th), .. each of the spaces in A, G, H, B, is th of a prime (or a second), and since there are seven from G to H, and three from H to B, .. the area of A, G, H, B, will be expressed by 7 x 3 21 seconds; but 12 seconds make a prime,.. 21 seconds are I prime and 9 seconds = area of A, G, H, B. 4. Since each of the spaces from P to E is 1 inch, and from P to M is one foot, .. each of the spaces in E, P, M, R, is a prime; but from P to G is 5 feet, .. E, P, G, A, will be expressed by 5 × 3 15 primes; or, 1 square foot and 3 primes area of E, P, G, A. But these several areas, viz., D, A, B, C; F, E, A, D ; A, G, H, B; and E, P, G, A, make up the whole area of the board; let them be added together, and compared with the usual rule.

=

=

The only difference between these two is, that instead of the 2 feet 4 inches (the area of the first part) being put down by themselves, and the area of the next, viz., 20 feet, put underneath, the 2 feet are carried in the right-hand example, and the feet become 22 instead of 20, as on the left.

In order to get another view of this subject, let us work with the equivalent fractions 5ths and 4ths.

Now this result is always called by workmen 23 feet, 8 inches, 9 seconds (understanding inches as square inches), an erroneous answer; and in expensive materials, as slabs of marble, the seller, calculating in this way, cheats himself considerably. It is recommended to the reader to ascertain the accurate and workmen's "cost in the above sum, if the dimensions are those of a polished marble slab, at 10s. 6d. per square foot.

It would improve the accuracy of the knowledge of this subject to draw a diagram, representing a board, which should be feet, inches, and twelfths long; and feet, inches, and twelfths, wide, and prove the rule in both of the above ways. This diagram would necessarily be larger than could be given in a magazine. It would be perceived by this, that the subdivisions are alternately rectangles and squares, and that although each subdivision is always th of some greater, yet the same area is sometimes a square, and sometimes a rectangle.

A table to be learnt, as feet x feet, give feet, &c., &c., is generally given, but this is unnecessary, for if the indices be added together, the proper denomination is given, ex. 2" x 3'-6".

Somers Town.

C. R. R.

NORMAL COLLEGE.

The following are the questions which have been proposed to the students in the Normal College at the recent Midsummer Examination.

The questions are divided into sections; the first question in each section is intended for the general class; the second for the first year students; and the third for those of the second year. No second question is to be attempted in any section till one question has been answered in every section.

ARITHMETIC AND ALGEBRA.

1. Enunciate the arithmetical truths which are assumed in the ordinary rule for working Long Division. Illustrate your answer by a complete analysis of the following sum:-Divide 796,384 by 58.

2. In what sense may the rules for Vulgar Fractions be regarded as extending our notions of Division? State some axioms which apply equally to both rules, and give two demonstrations of the proposition, that "to divide by a number is to multiply by its reciprocal."

3. What is the precise meaning of the sign (-) over the characteristic of a logarithm? Explain how, when the logarithms of any two numbers are known, those of any of the intermediate members can be deduced from them.

Given log. 21030103; find what would be the logarithm of 100 if the base were 8 instead of 10.

4. Find the price of 927 articles at 3 pounds 7 florins 8 cents 6 mils each, and prove your answer by showing how the same sum could be worked in the ordinary compound rule.

5. Show in what respects the system of decimal coinage, which takes £1 as the unit, and subdivides it into florins, cents, and mils, is preferable to one having a franc or tenpenny piece as its basis.

6. Name, as far as you are able, the countries in which a decimal system of money prevails; and give the units of value, with the respective equivalents in English coinage.

7. The poor's rate upon a house rated at 175 guineas is £6 17s. 9d.; what is the rate per pound, and how much should be paid in the same parish on a house whose annual rental is £120 ?

8. If it cost £27 18s. to supply a number of persons with bread at 74d. per 4lb. loaf, for what sum can two-thirds of that number be supplied when the price is 63d.?

9. Show that a number is divisible by 9 if the sum of its digits is divisible by 9; and find the smallest multiple of 9 which, when divided by 7, leaves a remainder 4.

10. Multiply x2 + ax + b by x2 — ax + c, and divide 1 -x by 1 + x --- x3.

12. The number of combinations of n things, taken r together, is the same as the number taken n-r together.

In how many ways can 17 balls be arranged in two divisions, containing respectively 12 and 5? 13. Solve the following equations:

(7) Two pipes fill a cistern in m and ʼn hours respectively; in what time will they fill it together?

Divide each of the numbers 11 and 17 into two parts, so that the product of the first parts of

each may be 45, and that of the second 48.

(n) 6x — x2 + 3 √ (x2 — 6x + 16) = 12.

ENGLISH LANGUAGE.

Treat the extracts below in the following ways:

(a) Parse, grammatically, the words in italics.
(b) Analyze, etymologically, the words in capitals.*
(c) Analyze, logically, the parts enclosed in brackets.

(d) Paraphrase the extract.

1. But some of Cicero's friends found means to give him early NOTICE of it, upon which he set forward presently, with his brother and nephew, towards Astura, the nearest villa, which he had upon the sea, with the INTENT to TRANSPORT themselves directly out of the reach of their ENEMIES. [But Quintus being wholly unprepared for so long a voyage, RFSOLVED to turn back with his son to Rome, in CONFIDENCE of lying concealed there till they could PROVIDE money and necessaries for their support abroad.]-Middleton's Life of Cicero.

2. In these four points consist, as I take it, the CONSTITUTIONAL NOTION of HEREDITARY right to the throne; which will be still further ELUCIDATED and inade clear beyond all DISPUTE from a short historical view of the SUCCESSIONS to the crown of England, the doctrines of our ANCIENT lawyers, and the several ACTS of Parliament that have, from time to time, been made, to create. to declare, to CONFIRM, to LIMIT, or to bar the hereditary title to the throne, [And in the PURSUIT of this inquiry, we shall find, that from the days of Egbert, the first sole MONARCH of this kingdom, even to the present, the four CARDINAL MAXIMS above mentioned have ever been held the cONSTITIONAL canons of SUCCESSION.]

3. [Thus while God spake, AMBROSIAL FRAGRANCE filled
All heaven, and in the blessed SPIRITS ELECT
SENSE of new joy INEFFABLE DIFFUSED.]
Beyond COMPARE the Son of God was seen,
Most glorious; in him all his Father shone
SUBSTANTIALLY expressed; and in his face
Divine COMPASSION VISIBLY appeared,
Love without end, and, without measure, grace;
Which, uttering thus, he to his Father spake.

O Father! gracious was that word which closed
Thy sovereign sentence, that Man should find GRACE;
For which both Heaven and Earth shall high EXTOL
Thy praises, with the INNUMERABLE Sound

Of hymns and SACRED Songs, wherewith thy throne
Encompassed shall resound thee ever blessed.

For should Man FINALLY be lost? Should Man,
Thy CREATURE late so lored, thy youngest Son,

Fall CIRCUMVENTED thus by fraud, though joined

With his own folly? That be from thee far.

That far be from thee, Father, who art judge

Of all things made, and judgest only right.

1. Give the third person singular of the verb to write in all the forms of the tenses of the indicative mood.

2. State the most important rules relating to verbal nouns.

3. Give two each of the most important syntactical rules of Agreement, Government, and Arrangement,

1. Give the etymologies of the following words:-artificial, secrete, indentation, interregnum.

2. Give, in addition to the above, the etymologies of the following:-antithetical, panorama, thoroughfare, polytechnic, nostril, prototype.

3. Give, in a table, a list of the terminations added to words of Anglo-Saxon, Latin, and Greek origin to make (a) verbs, (b) nouns denoting persons, (c) abstract nouns.

1. Show, by the use of a table, the various ways in which the subject, predicate, and object of a sentence may be enlarged.

2. Show, by the use of a table, all the different sorts of complex sentences.

3. Make a complex sentence, of which the subject of the principal sentence, and the predicate and object of the accessory sentence, shall be enlarged in as many ways as possible. Designate the sentence as a whole, and write over each adjunct its name in an abbreviated form.

1. Use the words that and all as different parts of speech in as many ways as you can.

2. Account for the case of the nouns and pronouns in the following sentences:-Give us this day our daily bread. Forgive us our trespasses.

3. Explain grammatically the following expressions: :-Woe worth the day. Earing and harvest.

Methinks.

GEOMETRY, MENSURATION, AND TRIGONOMETRY.

1. On the same base and on the same side of it, there cannot be two triangles, having the two sides terminated in one extremity of the base equal to one another, and likewise those terminated in the other extremity of the base; when the vertex of one of the triangles falls within the other. 2. Demonstrate one of the following propositions:

* The words in large capitals are to be parsed grammatically as well as analyzed etymologically.

8

3. The lines which bisect the angles of any parallelogram form a rectangular parallelogram whose diameters are parallel to the sides of the former.

4. What is an axiom? Enunciate all of those prefixed to the First Book of Euclid which have no special reference to geometry rather than to any other branch of mathematics.

5. "Propositions 9 and 10, in Book 11., are only differ ut forms of the same theorem." Explain this, and show in what other cases in the same book one truth is enunciated in two or more forms.

6. Describe the precise range and purpose of each of the first six books of Euclid.

7. Find the price of a piece of timber of which the length, breadth, and thickness are respectively 123 ft. 9 in.; 2 ft. 4 in.; and 2 ft., at 94d. per cubic foot.

8 By how much does the area of a circle whose circumference is 2,340 chains, differ from the united areas of two squares whose diagonals respectively are 187 and 259 chains?

9. How many acres are in a quadrilateral field, whose opposite angles are supplementary, the sides being 600, 650, 700, and 750 links, respectively?

10. State and explain the rules for finding-(a) the area of a triangle, when the base and perpendicular are given; (B) the area of a circle, when the circumference is given; (7) the cubic contents of a prism.

11. State and explain the rules for finding-(a) the third side of any triangle, when two sides and one angle are known; (3) the area of a triangle, when two sides and the contained angle are known; and (y) the base of a right angled triangle, when the hypotheneuse and one angle are known.

12. Express the radius of the circle inscribed in a triangle, in terms of the sides of the triangle. 13. Prove that if a and b be two sides opposite the angles A and B, then sin A: sin B: ab. Show what practical rules can be deduced from this proposition.

14 Express sine, cosine, tangent, and secant of 30°, 60°, and 45°, with numerical equivalents, and explain how these results are made available in the construction of tables.

3. The shores of the Atlantic, showing the course of the Gulf Stream.

1. Name all the towns of England having more than 100,000 inhabitants; give their exact physical position; and state briefly the most important particulars respecting them.

2. Excluding the capital, say which is the largest town in the following countries;-give the exact position, and other important particulars:-France; Austria; the Two Sicilies; Ireland; Prussia.

3. Give a brief, but not meagre, explanation of the phenomena of winds, particularly of those termed trade winds. Show why west winds prevail with us, and why they are generally warm. Name, also, the characteristic winds of various parts of the world.

1. Give the extent and population of the United Kingdom, as also of the whole British empire, which, in these particulars, compare with some other states.

2. Lay down, in a table, the colonies and dependencies of England, and give particulars respecting Malta, Gibraltar, and Jamaica.

3. Draw and describe the Nile or the Danube; the great desert of Africa, or that of Asia; the Andes, or the plains of South America.

1. Name and give particulars respecting the highest mountain and the largest lake in Europe, and the longest river in the west of England.

2. Name the most important exports of Prussia or Spain, and say what are the chief ports of the country you select.

3. What is the difference between the greatest meridian altitude of the sun, at the most south and the most north point of Europe.

1. Say where the following are, and for what noted:-Hastings; Stratford-on-Avon; Sca Fell; Naseby.

2. Give a brief description. physical and political, of Belgium or Italy.

3. (a) Say what you can of the geographical distribution of the following:--the cactus; the heaths; the bamboo; the palms; rice; the pouched animals; the bird of Paradise; the boa constrictor; and humming birds. (b) Name and give particulars respecting the characteristic animals and plants of Africa, Australia, or South America.

NATURAL PHILOSOPHY AND CHEMISTRY.

1. Define accurately the following terms :-aeroid, friction, weight, momentum, inertia, isochronous. 2. Give definitions and illustrations of the following terms:-centre of oscillation, coefficient of friction, atom, chemical equivalent, oxide, oxysalt, and chemical affinity.

3. Give formulæ for determining (a) the velocity generated by gravity in a given time, (B) the space traversed by a falling body in a given time, (7) the space traversed in the nth second of a body's descent, and (d) the height to which a body will rise if projected upwards with a given velocity.

4. Give formulæ for determining (a) the length of a pendulum when the time of its vibration is given, (B) the mechanical effect of rigidity in a rope, (y) the amount of friction between two surfaces, (d) the transverse strength of a beam of any given material, (e) the velocity of a jet of water issuing from an orifice, and (n) the pressure of water at any given depth below the surface.

5. Show how each of the three kinds of lever illustrates the general theorem concerning parallel forces. Determine the pressure on the fulcrum in each case, and give examples.

6. What are the best materials for resisting pressure, torsion, and transverse strain, respectively; and how far does the form of the material affect the result in either case.

7. Make a diagram of one of the undermentioned instruments, and explain the principle of its action-the steel-yard, the governor-ball, the capstan, the centrifugal pump, the wheel barometer. 8. Make aiagram and explain the action of one of the following:- the main spring of a watch, the screw-press, the pile-driving engine, the air-pump, the thermometer.

9. How do you estimate the total pressure of a liquid on the base and sides of a cubical vessel when filled? Why should a heavy body appear to lose weight when suspended in water, and how much does it lose?

10. "The surface of a fluid at rest is horizontal." Demonstrate this proposition.

11. Enumerate the more important elastic fluids, and arrange them, as far as you are able, in the order of their relative weight.

12. What is meant by latent heat? Describe an experiment by which its existence can be proved. 13. Enunciate the four laws of chemical combining proportion.

ANCIENT HISTORY.

1. Give some particulars respecting the life and and acts of the following persons :-Nabonadius, Darius Hystaspes, Homer, Miltiades, Solon, Herodotus, and Pericles.

2. Make a chronological table, as complete as you can, of the principal events in ancient history, to the death of Alexander.

3. Give a brief account of the second Persian invasion of Greece.

4. Name some important differences between the social and political condition of Athens, and that of Sparta. How may they be accounted for?

5. How long did the Peloponnesian war last? Who were the persons chiefly concerned in it? And what were the most important consequences which resulted from it?

6. Name the chief sources from which our knowledge of Greek history is derived.

ENGLISH HISTORY.

1. Give in a tabular form the following particulars respecting Henry II., Edward I., or Henry IV. :-The year he began to reign, the nature of his claim to the crown, and the most important events in his reign.

2. As above.

3. Give an account of the national industry during the Anglo-Saxon period, as regards (1) commerce, noticing especially the reign of Alfred, Athelstan, and Ethelred; (2) internal trade, and (3) the useful arts.

1. Up to the time of the Tudors, which of our kings came to an untimely end? Give some particulars respecting the same.

2. State what were the most important legislative enactments made by the Plantagenet kings who were most eminent as legislators.

3. In whose reign and in what way was the independence of the judges secured?

1. Name some of the most eminent persons who lived during the Anglo-Saxon period, and say for what they are noted.

2. Give some account of the Anglo-Saxon constitution, and give also the main features of the feudal system.

3. Give particulars respecting the buildings, furniture, and dress, during the period from the Norman Conquest to the death of John; and also concerning the social usages of that period.

1. Give some account of Dunstan, Becket, and John of Gaunt.

2. Show the nature of the claim to the crown in the cases of the following sovereigns, making a genealogical table where possible:-Richard II., Henry IV., Edward IV., Henry VII.