ANCYLOMETRY. 333 



parts of triangles ; by which, certain parts being given, the others 

 may be determined. Conic Sections, is the name applied to the study 

 of the curves formed by the intersections of a plane and a cone ; that 

 is, the circle, ellipse, parabola, and hyperbola. These curves are 

 often referred to, particularly in Astronomy and Navigation ; while 

 Trigonometry is also of frequent service, in these studies, and in the 

 practice of Surveying and Mensuration. 



Trigonometry, derives its name from the Greek r'ptyavoj, a triangle, 

 and fifepov, a measure. It is said to have been first investigated by 

 Hipparchus : but the oldest work extant upon it, is that of Menelaus 

 of Alexandria ; and the earliest trigonometrical tables which have 

 been preserved, are those of Ptolemy, in his Almagest. The Arabi- 

 ans simplified Trigonometry, by the introduction of sines, or the 

 half chords of double arcs, as the means of expressing angles : a 

 method employed in the writings of Albategnius, about A. D. 880 ; 

 though its invention is also claimed for the Hindoos. The Arabian 

 astronomer, Geber ben Aphla, in the llth century, compiled three or 

 four theorems, which became the basis of modern trigonometry. 

 MUller, of Germany, called also Regiomontanus, farther improved 

 Trigonometry by the use of tangents : and he was the first to resolve 

 spherical triangles, by finding the relations of their sides and angles. 

 To Napier, we are indebted, for his rules or Analogies, which assist 

 us in remembering the more difficult formulas ; and especially for the 

 invention of Logarithms, by which trigonometrical calculations are 

 so greatly simplified. Other improvements have been made by Euler 

 and others ; and the formulas of Trigonometry are now become so 

 general, and complete, as to leave but little more to be expected, or 

 even desired. 



The first examination of Conic Sections, has been attributed by 

 some writers to Menechmus, a friend of Plato ; and by others to 

 AristfEiis, whose writings are lost. The earliest work extant, on this 

 subject, is that of Apollonius of Perga, who flourished about 150 

 B. C. ; and who ranked next to Archimedes, as a geometer. Dr. 

 Wallis, in 1655, introduced the method of studying these curves 

 without any necessary reference to their being sections of a cone. 

 Galileo, Kepler, and Newton, discovered that the orbits of the planets 

 are curves of this class ; since which discovery they have been very 

 extensively and carefully studied. 



The invention of the modern Analytic Geometry, is attributed to 

 Vieta, and Descartes. Vieta applied it only to the construction of 

 the roots of equations ; but Descartes, in 1637, by the invention of 

 coordinates, found the means of designating geometrical curves by 

 algebraic equations ; in which the essence of this branch consists. 

 Descartes applied this system to curves of double curvature, by 

 means of their two projections ; but Maclaurin discovered a more 

 direct method, by means of triple coordinates, parallel to three 

 different axes, and related to each other by the nature of the curve, 

 or surface. It is only since the time of Descartes, that Trigonome- 

 try and Conic Sections have been treated analytically, and thus be- 

 come a part of this branch of Mathematics ; which has thus aided 

 the study of pure Geometry. 



