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the present treatife is certainly as much expofed as any other; for no attempt is here made to abridge the Elements, by confidering as felf-evident any thing that admits of being proved. Indeed, those who make the objection juft ftated, do not seem to have reflected fufficiently on the end of Mathematical Demonftration, which is not only to prove the truth of a certain propofition, but to fhew its neceffary connection with other propofitions, and its dependence on them. The truths of Geometry are all neceffarily connected with one another, and the fyftem of fuch truths can never be rightly explained, unless that connection be accurately traced, wherever it exifts. It is upon this that the beauty and peculiar excellence of the mathematical fciences depend: it is this, which, by preventing any one truth from being fingle and infulated, connects the different parts fo firmly, that they must all ftand, or all fall together. The demonftration, therefore, even of an 'obvious propofition, anfwers the purpose of connecting that propofition with others, and afcertaining its place in the general fyftem of mathematical truth. If, for example, it be alleged, that it is needless to demonftrate that any two fides of a triangle are greater than the third; it may be replied, that this is no doubt a truth, which, without proof, moft men will be inclined to admit; but, are we for that reafon to account it of no confequence to know what the propofitions are, which muft cease

to

to be true if this propofition were fuppofed to be falfe? Is it not useful to know, that unless it be true, that any two fides of a triangle are greater than the third, neither could it be true, that the greater fide of every triangle is oppofite to the greater angle, nor that the equal fides are oppofite to equal angles, nor, laftly, that things equal to the fame thing are equal to one another? By a fcientific mind this information will not be thought lightly of; and it is exactly that which we receive from EUCLID'S demonftration.

To all this it may be added, that the mind, efpecially when beginning to ftudy the art of reasoning, cannot be employed to greater advantage than in analysing those judgments, which, though they appear fimple, are in reality complex, and capable of being distinguished into parts. No progrefs in afcending higher can be expected, till a regular habit of demonftration is thus acquired; it is much to be feared, that he who has declined the trouble of tracing the connection between the propofition already quoted, and thofe that are more fimple, will not be very expert in tracing its connection with thofe that are more complex; and that, as he has not been careful in laying the foundation, he will never be fuccefsful in raifing the fuperftructure.

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ELEMENTS

OF

GEOMETR Y.

“A

BOOK I.

DEFINITIONS.

I.

POINT is that which has pofition, but not magni- See Notes. tude."

II.

A line is length without breadth.

"COROLLARY. The extremities of a line are points; and "the interfections of one line with another are also points."

III.

"Lines which cannot coincide in two points, without coin"ciding altogether, are called ftraight lines.

"COR. Hence two ftraight lines cannot inclose a space. Nei"ther can two ftraight lines have a common fegment; "that is, they cannot coincide in part, without coinciding "altogether."

IV..

A fuperficies is that which hath only length and breadth.
"COR. The extremities of a fuperficies are lines; and the in-
❝terfections of one fuperficies with another are alfo lines."

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A plane fuperficies is that in which any two points being taken, the itraight line between them lies wholly in that fuperficies.

VI.

A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the fame fraight line.

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N. B. When feveral angles are at one point B, any one of them is expreffed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the ftraight lines that contain the angle meet one another, is put between the other two letters, and one of these two is fomewhere upon one of thofe ftraight lines, and the other upon the other line: Thus the angle which is contained by the ftraight 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 expreffed by a letter placed at that point; as the angle at E.'

VII.

When a ftraight line ftanding on an-
other ftraight line makes the adja-
cent angles equal to one another,
each of the angles is called a right
angle; and the ftraight line which
ftands on the other is called a per-
pendicular to it.

VIII.

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