GEOMETRY. 



EOMETRY (from two Greek words signify- 

 ing the earth and to measure) is that branch 

 of mathematical science which is devoted to the 

 consideration of form and size, and may therefore 

 be said to be the best and surest guide to the 

 study of all sciences in which ideas of dimension 

 or space are involved. Almost all the knowledge 

 required by navigators, architects, surveyors, en- 

 gineers, and opticians, in their respective occu- 

 pations, is deduced from geometry and other 

 branches of mathematics. All works of art are 

 constructed according to the rules which geometry 

 involves ; and we find the same laws observed in 

 the works of nature. The study of mathematics, 

 generally, is also of great importance in cultivat- 

 ing habits of exact reasoning ; and in this respect 

 it forms a useful auxiliary to logic. 



It has been frequently asserted, though appar- 

 ently with little foundation, that geometry was 

 first cultivated in Egypt, in reference to the 

 measurement of the land. Thales of Miletus, who 

 lived about 600 B.C., is among the first concerning 

 whose attainments in mathematical knowledge 

 we have any authentic information. Plato made 

 several important discoveries in mathematics, 

 which he considered the chief of sciences. A 

 celebrated school, in which great improvement 

 was made in geometry, was established about 

 300 B.C. To this school the celebrated Euclid 

 belonged. In modern times, Kepler, Galileo, 

 Tacquet, Pascal, Descartes, Huygens of Holland, 

 our own Newton, Maclaurin, Lagrange, and many 

 others, have enlarged the bounds of mathematical 

 science, and have brought it to bear upon subjects 

 which, in former ages, were considered to be 

 beyond the grasp of the human mind. 



As improved by the labours of mathematicians, 

 geometrical science now includes the following 

 leading departments : Plane Geometry, Solid 

 and Spherical Geometry, Spherical Trigonom- 

 etry, the Projections of the Sphere, Perpendic- 

 ular Projection, Linear Perspective, and Conic 

 Sections. But to these main branches of the 

 science there are added Practical Mathematics, 

 which may be defined as an elaboration of the 

 abstract doctrines of general mathematics in 

 application to many matters of a practical nature 

 in the business of life. For example, among the 

 branches of Practical Mathematics we find Prac- 

 tical Geometry, Trigonometry, Measurement of 

 Heights and Distances, Levelling, Mensuration 

 of Surfaces, Mensuration of Solids, Land-sur- 

 veying, Calculations of Strength of Materials, 

 Gauging, Projectiles, Fortification, Astronomical 

 Problems, Navigation, Dialling, &c. 



It is proposed, in this and the following 

 number, to treat of Plane and Solid Geometry, 

 and to give the reader some insight into the 

 Mensuration of Lines, Surfaces, and Volumes, and 

 some of the more important constructions in 

 Practical Geometry. 



The truths or propositions of geometry are not 

 discovered by experience like those of physical 

 91 



science, but are proved or deduced from certain 

 definitions and axioms, the truth of which is at 

 once admitted. The usual text-book of this 

 science in Britain has long been the well-known 

 elements of Euclid ; but in the following pages, 

 which contain all the leading propositions in 

 geometry, an attempt has been made to present 

 the demonstrations in a somewhat simpler form 

 than those of Euclid, and at the same time to 

 preserve all the rigour and completeness de- 

 manded by the present state of the science. 



PLANE GEOMETRY. 

 DEFINITIONS. 



A solid is a limited portion of space. 



A surface is the boundary of a solid, or that 

 which separates it from surrounding space. 



A line is the limit or boundary of a surface. 



A point is the limit of a line. 



A solid has extension in all directions ; but for 

 the purpose of measuring its magnitude, it is con- 

 sidered as having three dimensions, called length, 

 breadth, and thickness. 



A surface has only two dimensions namely, 

 length and breadth. 



A line has only one dimension namely, length. 

 The intersection of two surfaces is a line. 



A point has no extension, and has therefore 

 neither length, breadth, nor thickness. The inter- 

 section of two lines is a point. 



If two lines be such that they cannot coincide 

 in two points without coinciding altogether, each 

 is said to be a straight line. Hence it follows that 

 only one straight line can pass through the same 

 two points, or a straight line is determined in 

 position by two points through which it passes. 

 It follows also that two straight lines cannot 

 inclose a surface. 



A plane surface, or a plane, is that in which, 

 any two points being taken, the line which joins 

 them lies wholly in that surface. 



A plane rectilineal angle is formed by two 

 straight lines meeting at a 

 point. Thus the two lines AB 

 and AC, meeting at A, form ^*\ 



the angle BAC, or the angle ^^ \ 

 CAB. The angle may be 

 regarded as the amount of 

 opening between any two positions of a straight 

 line considered as moving always in the same 

 plane round a fixed point. Thus the opening 

 between the two positions AB and AC of the same 

 line moving round A as on a pivot, is the angle A, 

 or the angle BAC, or CAB ; the angle being 

 named by a letter placed at the vertex, or by three 

 letters, that at the vertex being between the two 

 others. 



When the line AC, in revolving round A, arrives 

 at the position indicated in fig. 3. and such that 

 the angle CAB is equal to the angle CAD, each of 



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