Here we have as clear a history of the origin and changes of our vernacular, and as exact a grammatical analysis, as the combined powers of erudition and research possessed by Dr. Latham might have been expected to produce. Faithfully to have studied this Hand-book is to have acquired an intimate and certain understanding of the structure, the idiomatic forms, and anomalies of our modern Anglo-Saxon, with what is better still-the spirit and genius of good English composition. This is the second edition of this incomparable standard book.

English, Past and Present. By RICHARD CHEVENIX TRENCH, B.D. (John W. Parker and Son.) Five Lectures, exactly answering to the title. They are easy, popular, and most beautifully illustrative of the peculiarities, the beauties, and the noble archaisms of our language. A pleasant introduction to more severe study, under the guidance of Latham.

A third edition of DR. ROGET's Thesaurus of English Words and Phrases. (Longmans.) The words, and a large number of the phrases of the language, are thrown into classes, or categories, each class divided into sections, and each section consisting of groups, numbered from 1 to 1000. An alphabetical index of words directs to the group of which each word is a part, and thus presents, for choice in composition, a number, generally large, of kindred terms. The use of the Thesaurus is to aid in the acquisition of copiousness and accuracy in composition.

The fifth volume of The Museum of Science and Art, edited by DR. DIONYSIUS LARDNER, (Walton and Maberly,) is full of useful scientific information. The subjects are, The Steam-Engine, The Eye, The Atmosphere; and, under the head of "Common Things," Time, Pumps, Spectacles, and the Kaleidoscope. The illustrative engravings are very numerous; and these, with the quality of the instruction given at a wonderfully low rate, place the "Museum of Science and Art" among the first examples of cheap and useful literature.

CAPTAIN FISHBOURNE, Commander of H.M.S. "Hermes," on her visit to Nankin, when the rebels had lately got possession of that city, publishes his Impressions of China. (Seeley and Co.) The impressions are very erroneous; and this book comes into the market just twenty months too late. These impressions concerning the religion of Tae-ping-wang and his people are merely what were at first prevalent among the more credulous.

Essays selected from Contributions to the Edinburgh Review, by Henry Rogers. (Longmans.) The third and concluding volume

POETRY.-LESSONS.

of a collection of Essays that ranks deservedly high in the estimation of the religious and learned world.

POETRY.

SONNET.

TO AN INFANT-SCHOOL TEACHER.

"Meditate upon these things; give thyself wholly to them; that thy profiting may appear to all."-1 Tim. iv. 15.

WOULDST thou in grace and usefulness excel,
And thus promote the glory of the Lord,
And at His coming gain "a full reward ?"

Shun every charming path that leads to hell,

Throng'd by the gay, the thoughtless, and the vain,
Who find that flow'ry pleasures pierce with pain.

But let thy heart, O child of God! repel

Each fond delusion, and with ardour swell

To serve thy race, and gain His heavenly smile,
Who fondly folds the lambs for whom He died.

See that His "little ones," unlearn'd in guile,
Are with "the milk sincere" full well supplied;
Invoke His aid in all thy loving toil;
And make thy only boast in Jesus crucified!
St. Helen's.

J. Cook.

LESSONS.

GEOMETRY.-Proposition 7.-Problem.

To bisect a given finite right line (A B).

Solution. With the extremities (A B) of the given line as centres, and any equal radii, greater than half the line, describe arcs of circles intersecting above and below. The line (CD) joining the points of intersection will divide the given line into two equal parts (A E, BE).

Demonstration. Radii drawn to the points of intersection (CD) will complete two triangles (ACD, BCD) which have one side (CD) common, and the two remaining sides (A C, A D) in one respectively equal to the two in the other, being radii of equal circles: hence, by Proposition 5, the angles at either point of intersection are equal but these are the vertical angles of two smaller triangles (ACE, BCE), which have one side (CE) common, and the other two sides (A C, B C) equal as before: consequently, by Proposition 2, the bases (AE, BE), which are the parts of the given line, are equal.

It will be shown hereafter how a right line may be divided into any number of equal parts.

Proposition 8.-Problem.

To erect a perpendicular to a given right line (A B) at a given point (C) within it.

Solution. A circle with any radius, round the given point (C) as centre, will cut off equal portions (C D, C E) from the given line. On the sum of these portions (DE) describe an equilateral triangle (D E F), Proposition 1: the line (FC) joining its vertex with the given points is perpendicular to the given line.

E

D

Demonstration. Two triangles are formed which have one side (FC) common, two other sides (CD, CE), intercepts on the given line, equal, by construction, and the remaining sides (F D, F E) equal, being sides of the equilateral triangle: hence, by Proposition 5, the angles (F CD, FCE) which the joining line makes with the given line are equal; that is, the joining line is perpendicular to the given line, by Def. 10.

When the given point is at or near one end of the given line, this solution is inapplicable, unless the line can be conveniently produced. This difficulty will be met when the needful Theorems are established.

Proposition 9.-Problem.

To let fall a perpendicular on a given indefinite right line (A B) from a given point (C) without it.

Solution. Round the given point, as centre, describe a circle large enough to cut the given line in two points (DE). With these as centres, and radii greater than half their distance, describe equal circles intersecting in a point (F) remote from the given point (C): the line (CF) joining these is perpendicular to the given line.

A

Demonstration. Draw radii of the three circles to the points of intersection (DEF); these complete two larger triangles (DCF, ECF), which have one side (CF) common, two other sides (CD, CE) equal, being radii of equal circles; hence, by Proposition 5, the angles at the given point (C) are equal: but these are the vertical angles of two smaller triangles (DCG, ECG) which have one of the containing sides (CG) common, and the other pair (CD, CE) equal, as before : therefore, by Proposition 2, the angles (C GD, CGE) which the connecting line makes with the given line are equal; that is, these lines are at right angles.

When the line is limited in one direction, and the given point is nearly over the end of it, recourse must be had to another mode of solution, which shall be given hereafter.

Def. 40. If two angles are together equal to two right angles, each is called the supplement of the other.

Def. 41. If two angles are together equal to one right angle, each is called the complement of the other.

A. G.

ALGEBRA.-There are a few simple algebraical processes which it will be well for the student to commit to memory before leaving the more rudimentary details of the science. The following are the most important of these:

(m + n)2 = m2 + 2mn + n2 ;*

By actual multiplication, we find,

m + n

m + n

m2 + mn

mn + no

m2 + 2mn + n2.

And similarly with the rest.

or, in other words, the square of the sum of any two quantities is equal to the sum of their squares plus double their product.

And, in like manner (m—n)2= m2 — 2mn + n2;

that is to say, the square of the difference of any two quantities is equal to the sum of their squares minus double their product.

Geometry furnishes us with a parallel to both these cases; for, as Euclid tells us (II., 4), if a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts; and, were we to represent the line as a divided into two parts m, n, the algebraical expression for its square would be a3 = m2 + 2mn + n2; and a similar parallel to the second readily be imagined, by supposing m to equal a – n.

(m + n)2 case may

Again, (m+n) (m — n) = m3 — n2; or, the product of the sum and difference of any two quantities is equal to the difference of their squares. In like manner, (m2+n2) (m2 — n2) = m2 — n1, (m3 + n3) (m3 — n3) = m3 — no, &c.

It is well to bear in mind that,

m3 + n3 ÷ m + n = m2 — mn + n2

m3 – 3 : m-n=me tmntn

1÷1−x=1+x+x2 + x3 + &c., in infinitum; as may

The sum might be worked on for ever, it would never end. Let us substitute, however, 1 for x, and what is the result? 1÷1–1, that is, 1÷0=1+1+1+ &c., in infinitum, =∞* But, however philosophical the notion of nothing may seem to be, the notion of pure nonentity cannot enter into mathematical operations of this kind. Many pages would be required for even the inadequate discussion of this interesting subject: it must, therefore, be taken for granted that by 0 an indefinitely small quantity is to be understood; and it is evident that such a quantity would have to be added to itself an indefinite number of times before the result could equal unity.

* is the usual symbol for infinity.