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It has been objected to many of the writers on Elementary Geometry, and particularly to Euclid, that they have been at great pains to prove the truth of many simple propositions, which every body is ready to admit, without any demonstration, and that thus they take up the time, and fatigue the attention of the student, to no purpose. To this objection, if there be any force in it, the present treatise is certainly as much exposed as any other; for no attempt is here made to abridge the Elements, by considering as self-evident any thing that admits of being proved. Indeed those who make the objection just stated, do not seem to have reflected sufficiently on the end of Mathematical Demonstration, which is not only to prove the truth of a certain proposition, but to shew its necessary connection with other propositions, and its dependance on them. The truths of Geometry are all necessarily connected with one another, and the system of such truths can never be rightly explained, unless that connection be accurately traced, wherever it exists. It is upon this that the beauty and peculiar excellence of the mathematical sciences depend : it is this, which by preventing any one truth from being single and insulated, connects the different parts so firmly, that they must all stand, or all fall together. The demonstration, therefore, even of an obvious proposition, answers the purpose of connecting that pro
position with others, and ascertaining its place in the general system of mathematical truth. If, for example, it be alleged, that it is needless to demonstrate that any two sides of a triangle are greater than the third ; it
be replied, that this is no doubt a truth, which, without proof, most inen will be inclined to admit; but are we for that reason to account it of no consequence to know what the propositions are, which must cease to be true if this proposition were supposed to be false ? Is it not useful to know, that unless it be true, that any two sides of a triangle are greater than the third, neither could it be true, that the greater side of every triangle is opposite to the greater angle, nor that the equal sides are opposite to equal angles, nor, lastly, that things equal to the same thing are equal to one another? By a scientific mind this information will not be thought lightly of; and it is exactly that which we receive from Euclid's demonstration.
To all this it may be added, that the mind, especially when beginning to study the art of reasoning, cannot be employed to greater advantage than in analysing those judgments, which, though they appear simple, are in reality complex, and capable of being distinguished into parts. No progress in ascending higher can be expected, till a regular habit of demonstration
is thus acquired ; it is much to be feared, that he who 'has declined the trouble of tracing the connection between the proposition already quoted, and those that are more simple, will not be very expert in tracing its connection with those that are more complex; and that, as he has not been careful in laying the foundation, he will never be successful in raising the superstructure.
COLLEGE OF EDINBURGH,
Dec. 1, 1813,
II. A line is length without breadth. “ COROLLARY. The extremities of a line are points ; and the interos sections of one line with another are also points.”
III. “ If two lines are such that they cannot coincide in any two points,
“ without coinciding altogether, each of them is called a straight
" line. - Cor. Hence two straight lines cannot inclose
Neither “ two straight lines have a common segment; that is, they cannot “ coincide in part, without coinciding altogether."
IV. A superficies is that which has only length and breadth. 66 Cor. The extremities of a superficies are lines ; and the intersec“ tions of one superficies with another are also lines.”
V. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.
VI. A plane rectilineal angle is the inclination of two straight lines to one
another, which meet together, but are not in the same straight line.
* The definitions marked with inverted commas are different from those of Euclid.
C N. B. - When several angles are at one point B, any one of them * is expressed by three letters, of which the letter that is at the ver"tex of the angle, that is, at the point in which the straight lines that "contain the angle meet one another, is put between the other two « letters, and one of these two is somewhere apon one of those straight
lines, and the other upon the other line : Thus the angle which • is contained by the straight lines AB, CB, is named the angle * ABC, or CBA ; that which is contained by AB, BD is named the · angle ABD, or DBA ; and that which is contained by BD, CB is • called the angle DBC, or CBD ; but, if there be only one angle at a point, it may be expressed by a letter placed at that point; as the angle at E.
VII. When a straight line standing on an
other straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.
word area denotes the quantity of space contained in a figure, without any reference to the nature of the line or lines which bound it.