PROGRESS OF MATHEMATICS AND MECHANICS 241 



first six powers of a binomial given, and a special exponential 

 notation introduced. He shows that the celebrated classical 

 problems of trisecting a given angle and duplicating a cube involve 

 the solution of the cubic equation, and makes important discoveries 

 in the general theory of equations for example resolving poly- 

 nomials into linear factors and deriving from a given equation 

 other equations having roots which differ from those of the first by 

 a constant or by a given factor. He solves Apollonius' famous 

 problem of determining the circle tangent to three given circles, 

 and expresses TT by an infinite series. He devises systematic 

 methods for the solution of spherical triangles. 



DEVELOPMENT OF TRIGONOMETRY. Many circumstances com- 

 bined to promote the development of trigonometry at this period. 

 It was needed by the military engineer, the builder of roads, the 

 astronomer, the navigator, and the mapmaker whose work was 

 tributary to all of these. 



Rheticus (George Joachim, 1514r-1576), "the great computer 

 whose work has never been superseded," worked out a table of 

 natural sines for every 10 seconds to fifteen places of decimals. 

 We owe to him our familiar formulas for sin 1x and sin 3x. The 

 notation sin, tan, etc. and the determination of the area of a 

 spherical triangle date from about this time. To this period belong 

 also the very important work of Mercator on map-making and 

 the reform of the calendar by Pope Gregory XIII. 



MAP-MAKING. Mercator (Gerhard Kramer, 1512-1594) de- 

 voted himself in his home city, Louvain, to mathematical geogra- 

 phy, and gained his livelihood by making maps, globes and 

 astronomical instruments, combined in later life with teaching. 

 His great world map, completed in 1569, marks an epoch in 

 cartography. The first "Atlas" was published by his son in 1595. 

 He gives a mathematical analysis of the principles underlying the 

 projection of a spherical surface on a plane. 



'If/ he says, 'of the four relations subsisting between any two 

 places in respect to their mutual position, namely difference of latitude, 

 difference of longitude, direction and distance, only two are regarded, 

 the others also correspond exactly, and no error can be committed as 



