the present treatise is certainly as much exposed as any other ; for no attempt is here made to abridge the Elements, by considering as felf-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 propo . sitions, and its dependence 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 proposition 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 may be replied, that this is no doubt a truth, which, without proof, most men will be inclined to admit; but, are we for that reason to account it of no consequence to know what the propofitions are, which must cease


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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 fides 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, erpecially when beginning to study the art of reason ing, 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 muclı to be feared, thåt 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 neyer be successful in raising the superstructure.


Oct, 12, 1804.


Page 292. 1. 21 for 36.6 read 33.6

312, 14 cancel the last C

In the three last lines, for (B-C) read (C--B)
323. 4 for Niew read thewn
406. u for enumeration read enunciation

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: A tude."

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Point is that which has position, but not magni. See Notes.

A line is length without breadth.
66 COROLLARY. The extremities of a line are points; and
“ the intersections of one line with another are also points."

III. s6 Lines which cannot coincide in two points, without coin

ciding altogether, are called straight lines. “ Cor, Hence two straight lines cannot inclofe a fpace. Nei

" ther can two straight lines have a common segment; " that is, they cannot coincide in part, without coinciding altogether."

IV. A superficies is that which hath only length and breadth. 56 Cor. The extremities of a superficies are lines; and the in$6 tersections of one superficies with another are also lines.


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Book I,

A plane superficies is that in which any two points being ta.

ken, the itraight line between them lies wholly in that su.

A plane rectilineal angle is the inclination of two straight

lines to one another, which meet together, but are not in
the fame itraight linea




E B N. B. • When several angles are at one point B, any one of them is expreiled by three letters, of which the letter • that is at the vertex of the angle, that is, at the point in ' which the straight lines that contain the angle meet one an• other, is put between the other two letters, and one of these

two is somewhere upon one of those straight lines, and the • other upon the other line : Thus the angle which is con“ tained by the straight lines AB, CB, is named the angle ' ABC, or CBA; that which is contained by AB, BD iş • named the angle ABD, or DBA ; and that which is con• tained by BD, CB is called the angle DBC, or CBD; but, • if there be only one angle at a point, it

a point, it may be exprefled • by a letter placed at that point ; 23 the angle at E.'

When a straight line standing on an.

oiher Itraight line makes the adja-
cent 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 per-
pendicular to it.


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