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rect demonstrations of geometry depend on this general principle, that when magnitudes have been considered in all the possible states in which they can exist, one of which they must assume, and when it has been proved that they do not assume any of these except a particular one, it is a legitimate inference that they must assume that one. This mode of reasoning is exemplified by Euclid in the 19th proposition of his first book, in which the magnitudes are conceived to exist in three states, as, greater, equal, or less, and it is first proved that the one magnitude is not equal to the other, then that it is not less than the other, and it is therefore inferred that it must be greater. Many of the propositions which require an indirect proof may be so stated as to be susceptible of being demonstrated on this plan, and when it can be conveniently adopted it is the most satisfactory mode of proof. To follow it out in every case to its full extent would, however, lead to tedious repetitions. The ancient geometers, therefore, had recourse to a different method, which is termed a reductio ad absurdum, in which two states of the magnitudes only are brought into view. But it ought to be understood that this mode of proof is strictly applicable when the magnitudes can only assume two states, for if it be proved that they do not exist in one of these states, they must exist in the other. An instance of this occurs in the 5th proposition of Euclid's third book: "If two circles cut one another, they cannot have the same centre." Here the magnitudes can only assume two states, the circles must either have the same centre or they must not, and Euclid proves that the circles cannot have the same centre, and therefore infers that the centres must be different, and in such cases the inference is perfectly valid.
When the reductio ad absurdum is applied in cases where the magnitudes may assume three states, it is not equally satisfactory, because one of the states must be kept entirely out of view. Of this mode of proceeding, Euclid has given an example in the 6th proposition of his first book. He proposes to prove, that "If the angles at the base of a triangle are equal, the sides opposite to them are also equal." He names the sides opposite the equal angles AB, AC, magnitudes which may evidently assume three states, and he reasons thus:-If AB and AC are not equal they must be unequal, and if unequal, one of them must be greater than the other. AB is then supposed to be greater than AC, and a part BD is cut off from AB, and supposed equal to AC; and DC being joined, it is demonstrated, by means of the 4th proposition, that if the supposition were true, the triangle DBC would be. equal to ACB. But this conclusion, being contrary to the 9th axiom, is rejected as absurd, and it is then inferred that AB is not unequal to AC, or that they are equal to one another. That AB is not unequal to AC is certainly a just conclusion, and the argument in support of it has a logical form; but the mind of the student cannot perceive how such a conclusion can be drawn from the proof adduced, which tends rather to perplex him than to command his rational assent.
It appears questionable, therefore, whether the conclusion has been arrived at in a legitimate manner. The only inference which seems naturally to follow from the proof adduced is, that AB is not greater than AC. This inference, however, would immediately suggest that AB might exist in a different state, that of being less than AC, and it would then be requisite to prove also that it did not exist in the state of being
less, either by giving the demonstration at full length, or by stating that it might be demonstrated in the same manner as the former supposition, and after AB has been proved to be neither greater nor less than AC, the conclusion legitimately follows that AB is equal to AC. But Euclid draws the same conclusion after exhibiting only half the evidence necessary to support it; and in this particular circumstance consists the insufficiency of his demonstration. The reductio ad absurdum appears, therefore, in such cases to be merely an artifice to avoid repetition, by keeping one of the states which the magnitudes may assume entirely out of view; and by its introduction the ancients have in some degree sacrificed accuracy of reasoning to brevity and elegance. It is by no means wonderful, therefore, since half of the proof is suppressed, that this mode of demonstration has always been regarded by the young student as unsatisfactory, and not adducing evidence sufficient for conviction. When the reductio ad absurdum, therefore, is introduced in all cases where the magnitudes may assume three states, he ought to be informed. that it is only an abridgment of the former method introduced for the sake of brevity, and instructed how to supply that part of the proof which is wanting. It appears preferable, however, by adopting a different method, to remove every obstacle from the path of the learner which may obstruct his progress, especially when there is no necessity for its introduction.
These prefatory observations seemed requisite to impart to the reader a previous knowledge of the various alterations, and of the mode of demonstration pursued in these Elements, before he entered on the perusal of the propositions.
GEOMETRY is that branch of Mathematical Science which treats of magnitudes or quantities which are extended and measurable.
Magnitudes are of one, or two, or three dimensions, which are respectively termed lines, surfaces, and solids, the discovery and demonstration of the peculiar properties and relations of which are the principal objects of geometrical investigation.
Geometry treats of the figures of such magnitudes as well as of their extent.
Plane Geometry treats of lines and figures, all of which lie in the same plane; and the elementary part of it is still farther restricted to such propositions as for their demonstration.or solution require only the introduction of the straight line and the circle. In this treatise, therefore, all the lines and figures introduced are supposed to be all situated in the same plane.
Definitions are either explanations of the terms employed in geometry, or concise descriptions of the various magnitudes. about which it is conversant; and as constituting the primary basis on which it rests, they generally, and with great propriety, form the introduction to the study of the science.
1. A POINT is that which has position, but not magnitude. 2. A LINE is length without breadth.
The extremities of a line are points, and the intersections of one line with another are also points.
3. When one line only of the same kind can be drawn from one point to another it is called a straight line.
Cor. 1. Hence two straight lines can neither enclose a space nor have a common segment; for if a second straight line were drawn between the same points, it must coincide in every part and become one with the first.
Cor. 2. A straight line is the shortest distance between two points.
4. Any line that is not straight is called a BENT Line. it consist of straight lines, it is called a CROOKED LINE; if no part of it be a straight line, it is called a CURVED line.
Cor. Any number of bent lines can be drawn from one point to another.
5. A SURFACE is that which has only length and breadth. Cor. The extremities of a surface are lines, and the intersections of one surface with another are also lines.
6. A PLANE SURFACE is that in which any two points being taken, the straight line between them lies wholly in that surface.
Cor. Hence if two plane surfaces coincide in more than two points not in the same straight line, the whole of the one shall coincide with the whole of the other.
7. A SOLID is that which has length, breadth, and thickness. Cor. The extremities of a solid are surfaces.
8. A PLANE RECTILINEAL ANGLE is the opening between two straight lines which meet in a point, called the vertex of an angle, and the two lines are called its sides.
9. When a straight line standing on another straight line