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LESSONS.

GEOMETRY.-Definition 1. Geometry is the science which explains the properties of extension or magnitude, without regard to matter. The statement of any of its truths as a subject for argument is called a proposition. When a proposition exhibits some geometrical construction to be effected, it is a problem: when it proposes a truth or truths to be demonstrated, it is a theorem. Def. 2. Magnitudes are of four kinds: solids, surfaces, lines, and angles.

Def. 3. A solid has three dimensions,-length, breadth, and depth. Def. 4. The boundaries of a solid are surfaces or superficies: hence, a surface has only two dimensions,-length and breadth.

Def. 5. The extremities of a surface are lines: hence a line is length without breadth.

Def. 6. The terminations of a line are points: hence, a point is position, but not magnitude.

Def. 7. Lines which cannot have two points in common without being wholly coincident, are called straight or right lines.

A right line has been defined, "the shortest way between two points." This assumes that there is a shortest way, which would seem to require proof, and can be proved on principles which shall be laid down. It is from this property, that the right line terminating in two points is called the distance between those points.

Def. 8. If a right line cannot meet a surface in two points without lying wholly in it, that surface is plane.

Def. 9. A plane rectilineal angle is the degree of opening or divergence of two right lines which meet in a point called the vertex of the angle.

To form a correct conception of this species of magnitude, imagine one line to move from coincidence with another by revolving round their common extremity. The direction of the moving line, estimated from the centre of motion, is continually changing, and varies more and more from that of the fixed line, until it is at length diametrically opposite, when one line is the continuation of the other. At this juncture the line will manifestly have made half a revolution. Proceeding still in the same direction, it will reach its initial position of coincidence with the fixed line, after having completed one revolution. And, as the angular motion may thus proceed through revolution after revolution, without end, angular magnitude may be considered capable of indefinite increase. This view is essential to the application of geometry in the higher branches of mathematics.

With regard to the relative magnitudes of angles: two of these quantities are said to be equal, when, if one were laid or superposed on the other, the two lines including the former could, without altering their mutual inclination, be made to coincide with the two lines including the latter. From this it appears that, in order to the equality of two angles, it is not necessary that the lines forming them should be of the

same length. Now let two equal angles be
placed so that the vertices and two lines shall
be coincident, and the other two in the same
plane with the coincident lines and on opposite
sides. The two lines most divergent will now
form an angle double of one of the former angles. On the same
principle we can, in imagination, form an angle triple, quad-
ruple, &c., of any given one, an angle equivalent to the sum or
difference of two given ones, &c.

Def. 10. When one right line standing on another makes the adjacent angles equal, each of these angles is called a right angle, and each of the lines is said to be perpendicular to the other.

When, as in the previous illustration, the moving line has gone through a quarter revolution, it forms a right angle with the fixed line, and forms an equal angle with the continuation of that line beyond the vertex.

Def. 11. An obtuse angle is greater than a right angle.

Def. 12. An acute angle is less than a right angle. An angle is designated either by a letter placed at the vertex, or by three letters, of which the middle one is at the vertex, and the two others are anywhere along the sides. Thus the angle in the margin may be referred to either as the angle A, or as the angle BAC. The latter mode ble when several angles have the same vertex. diagram, three angles have the same vertex A: that formed by the lines AB, AC, and which is designated BAC; that formed by AC, AD, designated CAD; and, lastly, the one formed by AB, AD, which is the sum of the two former angles, and is written BAD.

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Def. 13. Parallel right lines are in the same plane, and do not meet, however far produced, both ways.

This is the usual definition of parallel lines. It assumes that there can be right lines in the same plane, which would never meet, though indefinitely produced in both directions. This is a truth which has been rigorously demonstrated, as shall be shown; but it seems hardly consistent with the rigour of geometrical reasoning to admit it here as a concealed axiom. It is to be observed, also, that the non-occursibility of two lines in the same plane, is not, in all cases, a criterion of parallelism. For example, a right line can be drawn which would never meet either branch of an hyperbole, however far produced; and yet, as they continually approach, the hyperbole and the right line, so related, are no where parallel. The subject of parallel lines shall be taken up at greater length hereafter.

A. G.

ALGEBRA.-Algebra expresses the relations of abstract quantities. Abstract quantities may be expressed in any arbitrary manner. The letters of the English alphabet are generally used: thus, we may add a and b together, or we may subtract one from the other, or multiply or divide one by the other, without giving them any fixed numerical value.

Certain signs are used to express the operations of addition, subtraction, multiplication, and division. There are other signs also. These must now be explained.

+(which is read plus) is the sign of addition: thus, instead of saying, a added to b, we say, a plus b, and write it, a+b; and instead of saying, 5a with 3a added, we say, 5a plus 3a, and write 5a+3a, and their sum is 8a.

- (which is read minus) is the sign of subtraction: thus, instead of saying, a with b taken from it, we say, a minus b, and write it, a-b: so, if we wish to subtract 3a from 5a, we write 5a-3a, and the result is 2a.

There are two combinations of the preceding signs, very useful but

not so frequently met with: ± (plus or minus) shows that the quantity to which it is prefixed may be either added or subtracted; thus, 7a3a is either 10a, the sum, or 4a, the difference, of these two quantities.

(minus or plus) shows that the quantity to which it is prefixed may be either subtracted or added: thus, 7a3a is either 4a or 10a.

> (which is read into) is the sign of multiplication: thus, in multiplying a by b, we write a x b; for as a and b represent quantities, they can of course be multiplied; if a, b, and c are to be multiplied, the process is shown thus, a xbx c. This sign, however, is frequently omitted, and a b c is precisely the same as a × 6 × c, just as 7a is the same as 7 times a, or 7 × a.

(which is read by) is the sign of division: a divided by b is

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The sign between two quantities shows that their difference is to be taken, it being immaterial which of them is the greater: thus, A b indicates the difference between the quantities a and b; and if a be greater than b, ab is equivalent to ab, if b be greater than a, ab is equivalent to b-a.

The sign = (which is read equals) shows that the quantities are equal to each other: thus, 7a+3a=10a, 7a-3a=4a, 7 × 3=21, and so on.

The sign > signifies that the former quantity is greater than the latter: thus, 7a >3a. But < signifies the reverse: as, 3a <7a.

Brackets, ( ), { }, [], show that the quantities included

between them are to be taken collectively, or as one quantity: they are very useful in the higher operations of algebra, but may be simply illustrated here: {3a+4a+6a}+8b=13a +8b.

The sign.. signifies therefore: . signifies because.

The signs,,, &c., /, are used to express the square root, the cube root, the biquadrate, &c., nth root respectively of the quantity before which they are placed: thus 4 signifies the square root of 4, that is to say 2; in like manner nth root of 64, where n may have any value: the value of n, 64 will be 8; if n be 3, say, 4.

8

is 2; and 2/64 is the for if we make 2 to be 64 is 64, that is to

These explanations presuppose some acquaintance with the principles of arithmetic; and, if they be studied during the month, some lessons in following Numbers will be easily understood.

M. L. R.

ASTRONOMICAL PHENOMENA.

JANUARY, 1855.

By A. GRAHAM, Esq., Markree Observatory, Collooney.

THE EARTH will be at its least distance from the Sun on the 1st, MARS on the 25th. MERCURY will be at his greatest distance on the 5th, VENUS on the 12th.

MARS and JUPITER may be seen very close together, near the horizon, in the south-west, shortly after sunset, on the 1st. VENUS will be close to JUPITER on the 18th, and MERCURY on the 25th: but these phenomena will occur too near the Sun to be seen by the naked eye.

MERCURY will be in superior conjunction with the Sun on the 20th, JUPITER on the 30th.

SATURN is the only one of the primary planets which will arrest the attention during this month. Situate amid a fine array of fixed stars, Aldebaran very near, the Pleiades to the right above, Orion to the left below, he shines conspicuous, infinitely heightened in interest by the revelations of the telescope. His rings, seen by us obliquely, are projected into concentric and similar ellipses, of which the outer axes, on the 1st, are 46′′ and 20" respectively.

RISING AND SETTING OF THE SUN, FOR THE PARALLELS OF THE BRITISH ISLANDS.

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Rises. Sets. Rises. h. m. h. m. h. m. 17 59 4 98 8 117 56 4 218 5 4 128 15 217 48 4 36.7 56 4 288 5 4 198 14 4 98 26 3 58 317 36 4 527 43 4 46,7 50 4 38 7 58 4 30 8

Sets. Rises. Sets. Rises. Sets. Rises. Sets.

h. m. h. m.

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h. m.

3 488 32

4

28 26

3 36 8 46
3 51 8

3 22

39

3 38

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SUN AND PLANETS AT GREENWICH.

MERCURY. VENUS. MARS. JUPITER. SATURN. URANUS.

SUN.

Rises. Sets.

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H. T. & J. Roche, Printers, 25, Hoxton-square, London.

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