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referred to in the subsequent reasoning - is, to mark its importance, printed in italics.

The conclusions thus distinguished, or the most important of them, if entered in the Copy-books' prepared for this purpose, and published in connexion with this work, will supply a kind of Analysis of Euclid, to those who have gone through the subject, but who wish at any time, as on the approach of an examination, to refresh their memory by a cursory re-perusal. To such, the reading of the several steps, and an inspection of the figure, will, in most cases, be sufficient to recall the complete proof to the mind, without the trouble of going over the entire proposition a second time; and this will be a pleasing and most improving exercise, and tend strongly to impress not only the proof itself, but also the principle of the proof, on the memory.

Although the text of Dr. Simson has been, in the main, adhered to in the present edition, alterations have been made wherever there appeared to be any obscurity in the language which could be removed by the introduction of a step, or the variation or transposition of a sentence.

As examples of such alteration, may be mentioned, the introduction of an additional figure in prop. 27, book i., and the use of a definite form of expression to mark the distinction in indirect demonstrations between a conclusion true in itself and one correctly deduced, but from an incorrect hypothesis. In the former case, the conclusion is expressed in the ordinary way; in the latter, the word 'assumed' is employed in the premiss, and the word 'must’ in the conclusion, to indicate that the reasoning proceeds on a false assumption, although the reasoning itself is correct. This distinction, it is believed, will be found of considerable practical importance in teaching Euclid to young students.

A. K. ISBISTER. STATIONERS' SCHOOL:

August 1, 1862.

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THE SCHOOL EUCLID.

BOOK I.

DEFINITIONS.

I. A POINT is that which hath no parts, or which hath no magnitude.

II. A line is length without breadth.

III. The extremities of a line are points.

IV.
A straight line is that which lies evenly between its extreme points.

V.
A superficies is that which hath only length and breadth.

VI.
The extremities of a superficies are lines.

VII. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.

VIII. • A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.'

IX. 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. S

A

D

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B

E

B

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 vertex of the angle, that is, at the point in which the straight lines containing the angle meet one another, 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 contained by the straight lines, AB, CB, is named the angle ABC, or CBA; that which is contained by AB, DB, is named the angle ABD, or DBA; and that which is contained by DB, 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.'

X. When a straight line standing on another 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.

XI. An obtuse angle is that which is greater than a right angle.

XII.

An acute angle is that which is less than a right angle.

XIII.
• A term or boundary is the extremity of any thing.'

XIV.
A figure is tbat which is enclosed by one or more boundaries.

XV. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another.

XVI.
And this point is called the centre of the circle.

XVII.

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A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

XVIII. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.

XIX. 'A segment of a circle is the figure contained by a straight line, and the circumference it cuts off.'

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XX.
Rectilineal figures are those which are contained by straight lines.

XXI.
Trilateral figures, or triangles, by three straight lines.

XXII. Quadrilateral, by four straight lines.

XXIII. Multilateral figures, or polygons, by more than four straight lines

XXIV.

Of three-sided figures, an equilateral triangle is that which has three equal sides.

XXV.

An isosceles triangle, is that which has only two sides equal.

XXVI.

A scalene triangle, is that which has three unequal sides.

XXVII

A right-angled triangle, is that which has a right angle

XXVIII.

An obtuse-angled triangle, is that which has an obtuse angle.

XXIX.

An acute-angled triangle, is that which has three acute · angles.

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