The co-efficient of the second term being 2, we take its half 1, and square it, which still gives 1: this quantity

=

being added to both sides of the equation, we have

z2 + 2x + 1 = 63 + 1 =

64;

and extracting the square root of both sides, we obtain

consequently

z + 1 = ± 8,

z = + 7, or = ~ — - 9,

either of which values of z will answer the conditions of the example; for taking + 7, its square is 49, which added to twice 7, give 63; and on the other hand taking the value-9, its square is 81, from which subtracting twice 918, the remainder is still 68, as before.

Had the question been stated so as that the given quantities were 22-63 2z, the result would have been simi lar, and the value of the unknown quantity z would have been found + 7, or 9, as in the procceding example. Example 2. Given æ2+ 8 = 6г, required the value of x.

4 × 4 = 16 + 8 = 24 = 6 × 4.

Here, by transposition, we get the equation

=

=-8; and adding to each side the square of 3, which is half the co-efficient 6, 9, we obtain another equation, x2 6x + 9 = 9—8 = 1; and the square roots of each side are x31, agreeably to what was already observed, that the root of the first side of the equation will have the

sign, because the sign of the term having the co-efficiént is -.

The limits of this work will not allow these observations on algebraic calculation to be further enlarged: the student must therefore be referred, for more ample information respecting the nature and uses of this branch of the science of numbers, to the various extensive treatises already published on the subject; and for the understanding of such works the foregoing instructions will, it is hoped, be found of considerable utility.

THE

MODERN PRECEPTOR.

CHAPTER V. ·

OF GEOMETRY.

GEOMETRY is that branch of science which treats of the nature and properties of Lines, Surfaces, and Solids. The name is formed from two Greek words, signifying to measure the earth, or, simply, land-measuring. The principles of geometry, like those of arithmetic, must have been coeval with human society: it is true, that in the ruder states of existence, the hunter, the fisher, the shepherd, have but little occasion for nice ascertainment of the bounds and extent of their respective ranges of country; yet still they must adopt some standard by which to apportion the space requisite for the maintenance of their families and tribes.

Hence the origin of geometry is, by the most ancient writers, assigned to periods far before their own times. Herodotus, the father of profane history, refers it to the time of Sesostris, king of Egypt, who intersected that country by numerous canals, and divided it amongst the inhabitants, according to fixed proportions. Others have attributed its rise to the necessity existing in Egypt, of making yearly surveys and allotments of the land, after the overflowings of the Nile, which levelled, or otherwise effaced, all boundaries and limits of distinction between the possessions of the several cultivators.

The Greek philosophers agree in tracing their knowledge

of geometry back to Egypt; but to themselves it owed great part of the advances it made to perfection. The writings of Euclid, in particular, have constantly possessed the highest reputation among geometricians; and his Elements of Geometry are, to this day considered as the fittest to be placed in the hands of those who wish to attain a rational acquaintance with this most useful, and most entertaining branch of knowledge. The business of the following short tract on geometry shall be, to give the young student such a taste and idea of the nature and advantages of this science, as may induce him to prosecute the study in a more extended and methodical manner, than can be attempted in this work. In the first place, he must understand the following explanation of certain terms, chiefly borrowed from the Greeks, but in constant use amongst geometricians of all countries.

An Axiom is a proposition or assertion of which the truth is at first sight so manifest, than no proof or demonstration can make it more evident for instance.

:

1st. Two quantities, whether they be lines, surfaces, or solids, which are both equal to a third quantity, are equal to one another: thus two pounds of silver, accurately weighed in the same scales, and with the same weight, must be equal the one to the other.

2d. The whole of any given quantity is greater than any part of it.

3d. The whole of any quantity is equal to the sum of all its component parts taken together.

4th. Only one straight line can be drawn between two points.

1 5th. Two magnitudes, whether they be lines, surfaces, or solids, which being applied to one another, perfectly coincide in every point, or which exactly fill the same

pace, are equal the one to the other.

Ch. All right angles are equal.

A Theorem is a proposition where some truth is to be proved by reasoning, that is, by demonstration; and means a thing to be shown.

A Problem is a proposition containing some questions which requires a solution, and means a thing to be done.

A Lemma is some truth employed in the demonstration of a theorem, or the solution of a problem. This, as well as a theorem and a problem, is comprehended under the general term, proposition.

A Corollary is some consequence drawn from one or more foregoing propositions.

A Scholium means a remark on some foregoing proposition, showing its application, extent, or connection with other propositions.

An Hypothesis is a supposition, or something taken for granted, in the statement otherwise called the enunciation, or in the demonstration of a proposition.

To render geometrical operations more concise, certain marks or signs have been adopted, of which those most geuerally employed are the following.

(=) Two parallel and horizontal lines signify equality : thus AB CD, means that the quantity or line designated by the letters AB, is equal to that expressed by CD.

(+) The St. George's cross, or two lines horizontal and perpendicular crossing each other, signify that the quantities between which this sign is placed, are to be added together; and the name is Plus, a Latin word, signifying more: thus A+B=C mean, that the two quantities represented by A and B, when added together, are equal to the quantity represented by C:-3+5-8.

(-) A single horizontal line between two quantities, signifies that the one is to be subtracted from the other; and it is named Minus, from a Latin word, signifying less thus A-B-C, or 8-5-3. These two signs may occur in the same expression of quantity, thus D+E-F= G, or