PROPOSITION 1.

The hypothesis, or the thing assumed in this proposition is, that one straight line meets or intersects another straight line and thus makes two adjacent angles. The point to be proved is, that the sum of these two angles is equal to two right-angles. The enunciation is particular, and the demonstration, direct. The object of the references is to show that the deductions we make from step to step, are true. The pupil in his demonstration, should repeat in full the principles thus referred to.

The point asserted respecting this particular diagram is proved to be true. If we construct a different figure in which two straight lines meet or intersect each other, a similar process of reasoning will show that the sum of the two adjacent angles in that case, is also equal to right-angles. Hence it is inferred, universally, that the sum of two adjacent angles formed by the intersection of any two straight lines, is equal to two right-angles.

PROPOSITION II.

In this proposition the hypothesis is, that there are two straight lines which have two points common; and the point to be proved is, that such lines form one continued straight line.

PROPOSITION V.

The placing of one figure upon another, or supposing them to be so placed, is called superposition.

The enunciation of this proposition is general; the hypothesis is, that the given triangle has two of its sides equal; and the point to be demonstrated is, that the angles opposite to these sides, are equal. The demonstration of this principle, given by Legendre, is much more concise and simple than that given by Euclid.

PROPOSITION XII.

This proposition is said to be the converse of the preceding proposition; that is, the hypothesis of the former is the point to b

proved in the latter, and the hypothesis in the latter is the point to be proved in the former.

The demonstration is indirect. By supposing its contrary, we are brought to the conclusion, that a part is equal to the whole, which is an absurdity.

The subject of parallel lines is confessedly one of the most difficult in the elements of geometry. Various methods have been devised to overcome these difficulties; but however ingenious and subtle their reasoning may be, it is the misfortune of most of them to be unsatisfactory. It is not our present object to attempt to substantiate or refute them; nor to enter into an abstract discussion of their comparative merits.

Among the various methods which have been adopted in different ages, the more prominent are the three following:

1. To form a new definition of parallel lines.

2. To assume a new axiom, more obvious than that of Euclid, respecting parallels.

3. To adopt a course of reasoning founded on the definition of parallels and the properties of other lines already demonstrated, and thus avoid the necessity of a new axiom.

Euclid's axiom alluded to is this:

"If a straight line meet two other straight lines, so as to make the two interior angles on the same side of it taken together less than two right-angles, these straight lines being continually produced, will at length meet on the side on which the angles are, that are less than two right-angles.”

The objection to this axiom is, that it assumes as self-evident, what in fact is not self-evident, but needs proof. Commentators upon Euclid's elements have very generally discarded it and introduced a new one. Clavius is said to have been the first who pur

sued this course.

With respect to the new axiom which has been substituted in its stead, that given by Ludlam seems to have met with the greatest favor. It is adopted by Playfair, and substantially by the Edinburgh Encyclopedia, &c. It is this:

"Two straight lines which intersect one another," (i. e. which are

drawn through the same point,)" cannot both be parallel to the same straight line."

This axiom forms a corollary to the twenty-second proposition of Legendre. In the Edinburgh Encyclopedia it is made the basis

of the demonstration of the following proposition:

If a straight line EF, meet two parallel lines AC, BD, the sum of the two interior angles CEF, EFD on the same side, will be equal

lines AC, GK, are drawn parallel to the given line BD; which is impossible. (Ax. above.) Therefore, If a straight line, &c.

From this proposition it is easy to prove if a straight line inters secting two other straight lines, makes the sum of the two interior angles on the same side less than two right-angles, these two lines will meet if produced. For, GK and BD must either meet or be parallel to each other: but they are not parallel, for if they were, the sum of the two interior angles GEF, BFE, would be equal to .two right-angles as shown above, which is contrary to the hypothesis; consequently, since they are not parallel, they must meet if produced. This being granted, the rest of the theory of parallels is easily demonstrated.

With regard to the other methods, it may be remarked that most of them involve the idea of infinity, or of the motion of lines, which may not vitiate the reasoning, yet it must be confessed, deprives them of that clearness and beauty which characterise other demonstrations in elementary geometry.

To Legendre belongs the honor of first delivering the doctrine of parallels without a new axiom, and independently of all considerations about infinity and motion.

He has done this in two ways. The method adopted in the text is

simple, and though founded on a property partially discovered by measurements on a figure made accurately, as he admits, is, nevertheless, sufficiently rigorous to carry a full conviction of the truth of the proposition, and, to most minds, is doubtless satisfactory. The other method he gave in his Notes, and must be admitted to be strictly rigorous, but is too abstruse for an elementary work. In the language of the late Prof. Playfair, "it is extremely ingenious, and proceeds on this very simple and undeniable axiom, that we cannot compare an angle and a line as to magnitude, or cannot have an equation of any sort between them; a truth which had long been received in geometry, but led only to negative consequences, till it fell into the hands of Legendre."

Those situated within the parallels, but on different sides of the intersecting line, as AGH, GHD; or BGH, GHC, are called alternate interior angles, or simply alternate angles.

Those which lie without the parallels and on different sides of the intersecting line, as BGE, CHF; or AGE, DHF, are called alternate exterior angles.

Finally, those which are on the same side of the intersecting line, one without, and the other within the parallels, as EGB, GHD; or EGA, GHC, are called opposite interior and exterior angles.

The principles contained in these two theorems, are given as corollaries in the original. But the fact that they require to be demonstrated, together with their importance, and the frequent reference made to them in the subsequent parts of geometry, seem to entitle them to a distinct place among the propositions.

BOOK II.

Def. 2.-By many writers the diagonal of a parallelogram and other quadrilateral figures, is called its diameter. It is more accurate and perspicuous, however, to restrict the use of the term diameter to those lines which pass through the centre of the circle, the sphere and other curvilineal figures; while the term diagonal is applied to quadrilateral, and other rectilineal figures.

The diameter of a circle divides it into two equal parts, which are called semicircles.

Def. 5.-When the radii are perpendicular to each other, the sector is called a quadrant. A quadrant, therefore, is half of a simicircle, or a fourth part of a whole circle.

In the subsequent pages, when reference is made from one proposition to another, Arabic figures will be used, the first denoting the number of the Proposition, the other the Book,

It appears most natural to measure a quantity by another quantity of the same species; and upon this principle it would be convenient to refer all angles to the right-angle; which, being made the unit of measure, an acute angle would be expressed by some number between 0 and 1; an obtuse angle by some number between 1 and 2. This mode of expressing angles would not, however, be the most convenient in practice. It has been found more simple to measure them by arcs of a circle, on account of the facility with which arcs can be made equal to given ares, and for various other reasons. all events, if the measurement of angles by arcs of a circle is in any degree indirect, it is still equally easy to obtain the direct and absolute measure by this method; since, on comparing the arc which serves as a measure to any angle, with the fourth part of the circumference, we find the ratio of the given angle to a right-angle, which is the absolute measure.

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