<;KO.MKTKV 



157 



know tin- distances of P lii>ni XX' and YY' 

 it \\e know Ml' an. I <>M. o.M is called the 

 . -sa, M 1' ilic ordinatcui' the |H>int I', and tin- t\v> 

 IHT are i-allcd the CM cirdinate* of I'. It in 

 I to denote ().\I and Ml' by .; and y. If the 

 jioint I' l>e supposed tn move in the plane affording 

 me law, a certain relation will exist hetweeil 

 -o ordinates : this relation expressed in an 

 ei|iiation will he the equation to the curve traced 

 out h\ I*. To take a simple example. Let the 

 law according to which 1' moves he that its 

 di-taiicc troni XX' shall always IM- doul)le its 

 distance I'roni YY' ; then the equation to the 

 cmve traced out l>y P will be y = 2x. If it be 

 leijiiired to draw the curve traced out by P, we may 

 ue any values for x, and from the equation 

 determine the corresponding values for y. If we 

 a-.>uine the values 1, 2, 3, &c. for x, the correspond- 

 ing values of y will l>e 2, 4, 6, &c. Determine then 

 i he points whose co-ordinates are 1 and 2, 2 and 4, 

 .; and 6, &c. ; these will be points on the curve. 

 It is not dillicult to discover that the curve is in 

 this instance a straight line. 



If the law according to which P moves in the 

 plane be that it shall always be at the same dis- 

 tance from a fixed point, we" have only to specify 

 the distance (say c), and the co-ordinates of the 

 fixed point (say a and 6), and we shall find the 

 equation which expresses this law to be 



(a; - a) 2 + (y - by = c". 



If the distance be c, and the fixed point be the 

 origin O whose coordinates are and 0, the 

 equation will be 



x 9 + y* = c\ 



The-,e last two equations are those of a circle. 



A- two co-ordinates are sufficient to determine a 

 point in a plane, so a plane curve described accord- 

 ing to a certain law will be represented by an 

 (|iiation between two variables, x and y; viz. 

 . y) = 0. It may be mentioned that equations 

 of the first degree represent straight lines, those of 

 the second degree represent some form of a conic 

 section, those of higher degrees represent curves 

 which in general take their name from the degree 

 of their equations. The position of a point in 

 space is fixed when its distances from three planes, 

 usually taken perpendicular to each other, are 

 known ; in other words, three co-ordinates x, y, z 

 determine a point in space. Hence, if a curved 

 surface is given in form and position, and we 

 can express algebraically one of its characteristic 

 properties, and obtain a relation F (x, y, z) = 

 between the co-ordinates of each of its points, this 

 equation is the equation of the surface ; and every 

 'Illation F (x, y, z) = 0, whose variables x, y, z 

 are the co-ordinates of a point referred to three 

 planes, perpendicular or oblique to each other, 

 represents some surface, the form of which depends 

 "ii .he way in which the variables are combined 

 with each other and with certain constant quan- 

 tities. 



The system of co-ordinates explained above is 

 called the Cartesian, from Descartes. There are 

 other systems, but a concise account of them would 

 be unintelligible. 



Of the history of geometry only the briefest 

 outline can be given here, and this outline must 

 be restricted mainly to pure geometry. Tradition 

 ascribes (and modern research tenets to con linn 

 rather than to invalidate the ascription) the origin 

 ot geometry to the Egyptians, who were compelled 

 to invent it in order to restore the landmarks 

 effaced by the inundation of the Nile, but our 

 knowledge of their attainments is meagre. From 

 a papyrus in the- British Museum written by 

 Ahmes, po^jhly about 1700 B.C., we infer that 

 the Egyptian- discussed only particular numerical 



problems, Much an the measurement* of certain 

 areas and solid*, and were little acquainted with 

 general theorems. The history of geometry, there- 

 fore, as a branch of science begins with Thale* of 

 Miletus (540 :>4'2 ii.i:.). The principal discover)' 

 attributed to him is the theorem that the sides of 

 mutually equiangular triangles are proportional. 

 After 1 hales came Pythagoras of Samoa (born 

 about 580 B.C.). It is dillicult to separate the 

 contributions which Pythagoras made to geometry 

 from those of his disciples, for everything was 

 ascribed to the master. The Pythagoreans ap]x?ar 

 to have been acquainted with most of the theorems 

 which form Euclid's first two Ixmks, with the 

 doctrine of proportion at least as applied to com- 

 mensurable magnitudes, with the construction of 

 the regular solids, and to have combined arithmetic 

 with geometry. The theorem of the three squares, 

 one of the most useful in the whole range of 

 geometry, is known as the theorem of Pythagoras. 

 Hippocrates of Chios, who reduced the problem of 

 the duplication of the cube to that of finding two 

 mean proportionals between two given straight 

 lines ; Archytas of Tarentum, who was the first to 

 duplicate the cul>e ; Eudoxus of Cnidus, the in- 

 ventor of the method of exhaustions and the 

 founder of the doctrine of proportion given in 

 Euclid's fifth book ; Mensechmus, the discoverer 

 of the three conic sections ; Deinostratus and 

 Nicomedes, the inventors of the quadratrix and 

 the conchoid ; and Arista?us, are the principal 

 predecessors of Euclid. To Euclid (about 300 B.C.) 

 is due the form in which elementary geometry 

 has been learnt for many centuries, and his treatise, 

 the Elements, seems to have completely superseded 

 all preceding writings on this subject. Those 

 books of this treatise which are concerned with 

 geometry are so well known that it is superfluous 

 to refer to their contents. Archimedes of Syracuse 

 (287-212 B.C.) is the greatest name in Greek 

 science. Besides his important contributions to 

 statics and hydrostatics, he wrote on the measure- 

 ment of the circle, on the quadrature of the 

 parabola, on the sphere and cylinder, on conoids 

 and spheroids, and on semi-regular polyhedrons. 

 Apollonius of Perga (260-200 B.C.) wrote on 

 several geometrical subjects, but the work which 

 procured him in his lifetime the title of ' the great 

 geometer,' was his treatise on the conic sections. 

 Ptolemy, author of the Almagest, Hero, and 

 Pappus are the last important geometers belonging 

 to the Alexandrian school. 



After the destruction of Alexandria (about 640 

 A.D.) the study of geometry underwent a long 

 eclipse. The Romans contributed nothing either 

 to geometrical or indeed to any kind of mathe- 

 matical discovery. The Hindus from the 6th to 

 the 12th century A.D. cultivated arithmetic, 

 algebra, and trigonometry, but in geometry they 

 produced nothing of any importance. A some- 

 what similar statement may be made regarding 

 the Arabs, but it ought to be remembered that 

 they translated the works of the great Greek 

 geometers, and it was through them that mathe- 

 matical science was in the 12th century intro- 

 duced into western Europe. From that time till 

 the close of the 16th century, though editions of 

 the Greek geometers were published and com- 

 mented on, little or no advance was made in 

 geometry comparable to what took place in other 

 branches of pure or applied mathematics. 



In the beginning of the 17th century Kepler 

 and Desargues laid the foundations of modern pure 

 geometry, the former by his enunciation of the 

 principle of continuity, and by his extension 

 of stereometry to solids of which the spheroids 

 and conoids of Archimedes were particular cases, 

 the latter by his introduction of the method of 



