PRELIMINARY REMARKS. 315 



or the delineation of objects geometrically, as it involves no other prin- 

 ciples than those of Elementary Geometry, and differs chiefly in the 

 mode of applying those principles, we would include in the same 

 branch, under the common name of Geometry. There remains only 

 the science of Fluxions, as it was named by Newton, or the Differen- 

 tial and Integral Calculus, as it has been named by the French mathe- 

 maticians, to complete the list of branches in this department. 



The History of Mathematics, may, we think, be referred chiefly 

 to that of its individual branches. The knowledge of the ancients, 

 in this department, was evidently far inferior to that of the moderns. 

 Although they reckoned by tens; a fact which is adduced, among 

 others, as proving the common origin of the nations thus reckoning; 

 yet they did not use, and probably were not acquainted with the deci- 

 mal notation which has so greatly simplified our modern Arithmetic. In 

 Elementary Geometry, and the Conic Sections deduced therefrom, 

 the knowledge of the Greeks would bear a comparison with that of 

 modern times ; but in these branches only, of the Pure Mathematics. 

 Some of their most learned works were destroyed in the Alexandrian 

 Library, or during the dark ages ; but others were preserved by the 

 Arabians themselves, when a milder dynasty succeeded ; and the 

 Greek works collected and translated into Arabic, by order of the 

 Caliph Al Mamun, have supplied much of the information which we 

 now possess, concerning ancient science, (p. 289.) 



To the Arabians, we are indebted, for the introduction of the De- 

 cimal Notation, and for the science of Algebra ; which they appear 

 to have transmitted rather than invented ; as we shall have occasion 

 to show, in treating of the individual branches. Their mathematics, 

 being introduced by the Moors into Spain, was zealously cultivated 

 by Alphonso of Castile ; and from thence it was introduced into 

 France, as early as A. D. 970, by Gerbert, who afterwards became 

 Pope Sylvester II. It was disseminated in Italy, about A. D. 1228, 

 by Camillus Leonard, a rich merchant of Pisa, who had travelled in 

 the East; and at about the same time, John of Halifax, or Sacro- 

 bosco, of England, wrote a treatise on the Arithmetic of the Arabs. 

 From that period to the present, the progress of mathematics has 

 been continuous ; and the greatest natitms of Europe have been com- 

 petitors for the honor of its new discoveries and inventions. 



The invention of Analytic Geometry, by Descartes, and of Coor- 

 dinates, by Maclaurin, has greatly extended our means of investigating 

 curves, and curved surfaces in general, as well as their included solids. 

 The invention of Logarithms, by Napier, has simplified, in a won- 

 derful degree, the higher numerical calculations, which before were 

 extremely tedious. The invention of Descriptive Geometry, by 

 Monge, has given us a complete method of representing and mea- 

 suring geometrical magnitudes, and forms ; the applications of which 

 are of great practical value. And especially, the invention of Flux- 

 ions, or the Calculus, almost simultaneously by Newton and Leibnitz, 

 has opened the way to a new and wide range of mathematical investiga- 

 tion, quite beyond the reach of ancient science, and which has served, 

 in skilful hands, to detect and explain various laws of nature that 

 before seemed absurd or contradictory. 



