GEOMETRY 



2444 



GEOMETRY 



212 B. c. His work was rather in solid than 

 in plane geometry, and he had left a request, 

 which was faithfully carried out, that there 

 be cut on his tomb a sphere inscribed in a 

 cylinder, because the working out of that prob- 

 lem he felt was his greatest achievement. All 

 through the Middle Ages there was little in- 

 terest in geometry, and though this was awak- 

 ened at the time of the Renaissance no real 

 advance was made in the science until within 

 the last two centuries. It is useless to discuss 

 in such an article as this these more recent 

 developments, as they all relate to the more 

 abstruse phases of the subject. Suffice it to 

 say that the problems over which the school- 

 boy nowadays puzzles his brain are the very 

 problems which interested Euclid. 



What Geometry Concerns Itself With. The 

 above discussion of the way in which geometry 

 has developed does not really tell what it is. 

 Perhaps the simplest definition is to say that 

 geometry is the "science of space;" that is, it 

 concerns itself with the relations and proper- 

 ties of points, which have position but not 

 magnitude; lines, which have but one dimen- 

 sion, length; surfaces, which have two di- 

 mensions, length and breadth; and solids, 

 which have three dimensions, length, breadth 

 and thickness. This may sound at first rather 

 theoretical, but geometry is an eminently 

 practical subject. If a man wishes to bisect 

 (cut in two equal parts) a given line or a given 

 angle; to inscribe one figure in another; to 

 drop a perpendicular to a line from a point 

 without the line; to draw a line parallel to 

 a given line through a given point; to make 

 an angle equal to a given angle to do any 

 of these or a thousand other things which not 

 only the architect, the stonecutter or the 

 draughtsman, but the carpenter or the amateur 

 craftsman may be called upon to do at any 

 time, only geometry can help him to perform 

 his task simply and accurately. Indeed, many 

 of the problems which geometry has solved 

 have grown out of just such popular needs. 



Branches of Geometry. The geometry taught 

 in secondary schools is called elementary, no 

 matter how difficult and "advanced" it may 

 seem to the student who is becoming ac- 

 quainted with it for the first time; and ele- 

 mentary geometry is divided into two branches 

 plane and solid geometry. Plane geometry 

 treats of all figures which lie in a plane that 

 is, which have not more than the two dimen- 

 sions of length and breadth and have no part 

 of their surface curved; while solid geometry 



treats of simple solids, or objects having the 

 three dimensions of length, breadth and thick- 

 . ness, such as cylinders, cubes, spheres or cones. 

 With any other curve than that of the circle, 

 or with plane surfaces or solids bounded by 

 any other curves, elementary geometry has 

 nothing to do. 



It was elementary geometry almost entirely 

 to which the ancients who built up the science 

 devoted themselves, but modern mathemati- 

 cians have worked out two great systems in 

 addition analytical geometry and projective 

 geometry. 



Analytical geometry is in reality a sort of 

 combination of geometry and algebra. The 

 relations of geometric figures are expressed in 

 algebraic terms, equations are formed and 

 worked out just as in algebra, and the result- 

 ant equation is translated back into geometric 

 figures. This makes possible the study of 

 more complex curves than elementary geom- 

 etry can deal with. First of all, the laws of 

 conic sections may be understood by means of 

 this algebra-geometry process. If a cone be 

 cut in a plane parallel with its base, the cut 

 surface is a circle, but if it be cut in any other 

 way the surface resulting is bounded by less 

 simple curves; and these less simple curves, 

 or conic sections, as they are called, are favor- 

 ite subjects of analytical geometry. Spirals, 

 too, and wavy lines, cannot conceal their prop- 

 erties and their relations from the person who 

 has mastered this higher form of geometry. 

 The problems worked out are all too difficult 

 for solution here, but it is interesting for even 

 the beginner to know that such an all-embrac- 

 ing form of geometry does exist. 



Projective geometry is abstruse enough in 

 its working out, but in its principles it is sim- 

 ple. It studies not figures themselves, but 

 their projections. Does that sound difficult? 

 If, walking on a sunny day with the sun be- 

 hind you, you look downward, you will see 

 there your shadow, long or short according to 

 the height of the sun. That shadow is your 

 projection, and it is like you in some ways 

 it has a head and two hands and two feet; but 

 it is not by any means a duplicate of your 

 figure. Now there are certain laws which gov- 

 ern projections and their relation to the fig- 

 ure from which they are projected, and with 

 these laws projective geometry concerns itself. 

 Old maps are very quaint and hard to under- 

 stand, because the ancients did not know any- 

 thing about the projection of a sphere on a 

 flat surface, which is exactly what map-draw- 



