Cunha, and which, as Professor Playfair remarked, in the Edinburgh Review, vol. XX., is a decided improvement in elementary geometry, as it dispenses with an awkward subsidiary proposition of Euclid.

Upon the doctrine of proportion, which constitutes the fifth book of these elements, I have bestowed much labour and attention, and have, I hope, in some degree succeeded in diminishing the difficulties hitherto attendant upon that important subject.

The notes appended to this first part may, I think, be consulted by the student with advantage. I have therein endeavoured to point out some remarkable errors and inconsistencies into which modern geometers have fallen, particularly in reference to the theory of parallel lines, and the doctrine of proportion; and I believe many of these errors have hitherto remained unnoticed. A singular instance of this is shown in the notes to the sixth book, where a proposition in Simpson's Geometry, which has been for upwards of seventy years received as genuine, and adopted by more modern geometers, is proved to be false! Other instances of incautious reasoning are adduced from Legendre, Dr. Simson, and others, which it is doubtless of importance to detect and point out to the student, as indisputable proofs of the great caution necessary in geometrical reasoning.

Throughout the whole I have earnestly endeavoured to render this performance suitable to the wants of the student, and deserving of the approbation of the geometer. I can truly say that its composition has been attended with a great sacrifice, both of labour and ex

pense; and its progress has been frequently interrupted by opposing circumstances. But if, notwithstanding, I shall have succeeded in rendering it worthy of notice, I shall consider myself fully recompensed for the pains it has cost me, and shall feel encouraged to proceed with more confidence and ardour in the remaining part of the subject.

June 1st, 1827.

J. R. YOUNG.

The second part will contain the GEOMETRY OF PLANES AND SOLIDS, with notes and an appendix on the SYMMETRICAL POLYEDRONS of Legendre.

The student is recommended to correct the following errors with a pen, particularly those which occur in pages 72 and 76.

ERRATA.

Page 1, bottom line, for with, read to.

15, line 2 from bottom, for bAC, read bAc.

16, line 9, for C, read B.

72, lines 6 and 7, for R>S, read R> S'.

ib. for PQ in line 6, read P > Q'.

ib. in lines 9 and 15, for Q', S, read Q', S'.

76, lines 10 and 12, for oftener, read more or less often.

78, line 13, for three, read four.

91, line 6 from bottom, for C, read E.

108, supply the line GE in the diagram.

109, line 18 from bottom, for AB2+ AC2, read AB∙AC. 131, line 5, for Oad, read OaD.

ELEMENTS OF GEOMETRY.

BOOK I.

GEOMETRY is the science which treats of the properties, relations, and measurement of magnitude in general. Magnitude can have but three dimensions, length, breadth, and thickness, all of which are necessary to constitute a body, or solid. It is important, however, to consider magnitude under the three distinct denominations of lines, surfaces, and solids, and thus the science of Geometry becomes divided into three principal branches: the first part treating of lines described upon the same plane, and of the surfaces which they enclose; the second of lines situated in different planes, and of the relations of these planes to each other; and the third part contemplating body under its several dimensions of length, breadth, and thickness. Lines are obviously the boundaries of surfaces, and surfaces are the boundaries of solids: it is equally obvious that a line, being mere length, without either breadth or thickness, can exist only as the boundary of a surface, and that a surface being absolutely without thickness, can exist only as an attribute of body. Although, therefore, it cannot be supposed that a line, or a surface, can have separate or independent existence, the fact will not in the smallest degree interrupt or embarrass our reasonings in considering these several attributes of body or space, each apart from the others, nothing more being requisite than the abstracting these others from our inquiry; so that in considering lines, length only is recognized, and in contemplating surfaces, length and breadth are combined, and thickness excluded. Having made these preliminary remarks, which were deemed essential to the student, we may proceed with the definitions.

B

DEFINITIONS.

1. Straight lines are those of which but one can be drawn from one point to another.

Two straight lines, therefore, cannot include space.

2. When two straight lines meet, the opening between them is called an angle; the point of meeting is called the vertex, and the lines themselves, which are said to contain the angle, are called the sides of the angle.

An angle is referred to simply, by means of the letter, at its vertex. Thus the angle contained by the straight lines AB, BC, is designated as the angle B. When, however, two or more

B

angles have the same vertex, then, in order to denote any one in particular, it becomes necessary to specify its sides by employing the three letters at their extremities; that at the vertex being always placed in the middle.

Thus the angle CBA or ABC, denotes that particular angle having the vertex Band contained by the sides AB, CB, and by the angle DBC or CBD, is in like manner meant the angle whose vertex is B, and whose sides are BD, BC.

A

E

B

D

It is obvious that the quantity of an angle depends not upon the length, but entirely upon the position of its containing sides; for the opening between the sides AB, CB, must remain the same, however these lines be increased or diminished.

3. One straight line is said to be perpendicular to another, when it makes with it equal adjacent angles. A perpendicular at the extremity of a line, is that which makes an angle with it equal to the adjacent angle, which would be formed by prolonging the line beyond that extremity.

4. A right angle is the angle formed by a straight line and a perpendicular to it.

5. An acute angle is less than a right angle.

6. An obtuse angle is greater than a right angle.

7. A plane surface, or simply a plane, is that in which, if any two points whatever be taken, the straight line which joins them will lie wholly in it.

8. A straight line is said to be parallel to another when they are in the same plane, and can never meet, however far they may be produced.

If, for example, the straight line AB, how far soever it be produced, can never meet the prolongation of CD, which is in the same plane, it is said to be parallel to it.

A

C.

B

9. By the distance of a point from a straight line, is meant the perpendicular from that point to the line; and one line is said to be equi-distant from another, when every point therein is equally distant from it,

10. A plane figure is an enclosed plane surface.

11. If it be bounded by straight lines only, it is called a rectilineal figure.

12. A polygon is a name used to comprehend every rectilineal figure, without regard to the number of its sides. The boundary of the figure is called its perimeter.

13. Among polygons, however, are more particularly distinguished the figure of three sides, called a triangle, and that of four sides called a quadrilateral.

14. An isosceles triangle is one, which has two equal sides.

15. An equilateral triangle is one which has all its sides equal.

16. When no two sides are equal the triangle is called scalene.

A