396 THE SCIENTIFIC MONTHLY 



supremacy, but in each of these civilizations we see signs of decline in 

 a far shorter period. At present there appears to be no indication that 

 the crest of the wave of scientific progress has been reached. The world 

 war just concluded, in its essence a struggle for liberty of thought and 

 action, was far too short to submerge the knowledge of the existing 

 generation and prevent it from being handed on to the next. 



The period can be roughly divided into two parts, the two centuries 

 until the time of Newton forming the first of these. It is chiefly 

 characterised by a sustained effort to lay solid foundations for all those 

 sciences in which mathematics is needed for development, and success 

 very nearly complete was attained at the end of the seventeenth century. 

 In Astronomy and mechanics, Copernicus placed the sun in the center 

 of our solar system. Kepler, building on the observations of Tycho 

 Brahe, gave the laws of the planetary motions. Galileo laid founda- 

 tions for the study of mechanics and proved the rotation of the earth 

 about its axis, besides making a host of astronomical discoveries with 

 his telescope. In mathematics, the arabic numerals came into full 

 use and were applied to the needs of daily life. The symbolism of 

 algebra was completed in nearly the modern form used in elementary 

 mathematics: much of the work done by Vieta, Cardan and Tartaglia 

 involved the solution of equations of the third and fourth degrees. 

 Trigonometry, needed by the astronomer, the map-maker, and the navi- 

 gator, was developed so far that Rheticus calculated a table of natural 

 sines to every 10 seconds of arc and to 15 places of decimals. 

 Logarithms were invented by Napier, and seem to have been adopted 

 universally almost immediately. Descartes invented analytical geom- 

 etry and thus gave to the investigators of space-forms a new weapon 

 of immense power, while Desargues laid the foundation of projective 

 geometry. Fermat, an amateur and perhaps as able as any of his con- 

 temporaries, laid down many of the laws of numbers and founded the 

 calculus of probabilities. 



The brevity of this summary is not a measure of the achievements 

 of these two centuries- For this we must look to those which followed. 

 While judged by modern standards, the actual progress seems small, 

 it was far beyond that which had been achieved in all the previous ages. 

 The only earlier contribution which has stood the test of time is per- 

 haps the geometry of the Greeks, for the work of Archimedes, especially 

 in mechanics, seems to have remained almost unknown up to the time 

 of Galileo. The period showed not only a sound laying of founda- 

 tions, but in its development of ideas, often only dimly expressed, 

 showed that the germination of the seeds of future knowledge had 

 already begun. Progress was frequently hampered by an unfortunate 

 form of rivalry in which a discovery was kept back so that its possessor 

 could propose problems to confute the claims for knowledge of his 



