284 Cambridge Course of Mathematics. 



Calculus; and various academical memoirs. In the late 

 great trigonometrical surveys in France and England, op- 

 erations which have contributed so much to our knowl- 

 edge of the earth we inhabit, and to the honor of the na- 

 tions engaged in them ; the instruments used were of such 

 exquisite construction, that the angles were measured to a 

 fraction of a second. Hence, the spherical excess, that is, 

 the excess of the three angles of the triangles, measured in 

 these surveys on the surface of the earth, above two right 

 angles, became apparent. It was necessary, therefore, to 

 estimate this excess. The prompt genius of M. Legcndre 

 furnished for the occasion a theorem, founded on the fact, 

 that the spherical triangles whose angles are measured in 

 trigonometrical surveys, have their sides very small when 

 compared with the radius of the sphere, which is, in this 

 case, the radius of the earth. The theorem adverted to, 

 reduces the resolution of these spherical triangles, to that of 

 rectilineal triangles, and admirably unites conciseness with 

 a sufficient degree of exactness. The theorem is this: '^^ 

 spherical triangle being proposed, qfzohich the sides are very 

 small in relation to the radius of the sphere, if from each of 

 its angles one third of the excess of the sum of its three an- 

 gles above two right angles, be subtracted, the angles so di- 

 minished, may be taken for the angles of a rectilineal trian- 

 gle, the sides of which are equal in length to those of the pro- 

 posed spherical triangle.''^ M. Legend re has given a den^.on- 

 stration of this very valuable theorem, in the appendix to 

 his Trigonometry, 



The Elements of Geometry now under consideration, 

 were composed during that period of the F'rench history, 

 when the ancient foundations of society and government 

 were undermined, and the political edifice throughout Eu- 

 rope, rocked with fearful convulsions on its base. The 

 great French philosophers and mathematicians saw plainly, 

 that in order to conciliate the popular favor, and avoid the 

 jealousy of the reigning authority, they must, as far as pos- 

 sible, render their favorite pursuits subservient to objects of 

 immediate and practical utility. They had seen the emi- 

 nent talents and conspicuous virtues of Lavoisier, insuffi- 

 cient to save him from the scaffold ; and the illustrious but 

 unfortunate Bailly, who had formerly been the idol of the 

 French nation, and who had devoted his life to the inter- 



