700 GEOMETRY compasses. If the solution of a problem re- quires in its graphic representation a line which cannot be drawn by means of these two in- struments, it was not considered by the an- cient Greek geometers a geometrical solution. Elementary geometry is sometimes subdivided into planimetry and stereometry, the former treating only of such lines and figures as lie in a plane, the latter of solids bounded by planes, and of the sphere, cone, and cylinder, which are usually designated as the three round bodies. That part of planimetry which treats of the measurement of triangles, and shows how, the magnitude of certain parts of a triangle being given, the magnitude of the other parts can be found, is called trigonome- try, and, on account of the peculiarity of the methods which it employs, is usually treated as a separate branch of geometry. Geometry again is divided into synthetic and analytic, or ancient and modern, or special and gene- ral ; divisions which all signify the same thing, and are based upon the difference between the methods which are employed in them respec- tively. Synthetic, ancient, or special geometry is founded upon the direct observation of the forms or figures themselves, and all its reason- ings are conducted with direct reference to those figures. Thus, in treating the ellipse, the first thing to be done, according to this method, is to draw an ellipse upon a plane, or to draw a representation of a cone with a plane passing through it obliquely to its axis, or in any other convenient way to bring before the mind the actual figure; next, to draw such other lines as the course of the reasoning may require ; and lastly, to demonstrate, in ac- cordance with the rules of logic and previously established propositions, the different propor- tions of the figure. This method, when com- pared with the analytic, modern, or general method, of which we shall presently speak, possesses great advantages and disadvantages. Among the former we mention that it is evi- dently the natural method, that is, the method to which the human mind naturally would and must resort in its first attempts to investigate the relations of space. It keeps not only be- fore the mind but before the eyes the actual thing whose nature we are investigating, and constantly calls upon the hands to do what the mind has conceived. As a mental disci- pline, geometry studied in this manner is not surpassed, perhaps not equalled, by any other science. Especially in the solution oif prob- lems reason, ingenuity, and imagination are all called into exercise. This method was the only one known to the ancient Greeks, and they regarded geometry as holding the highest rank among the sciences. Plato is said to have inscribed over his door, "Let no one en- ter here who is unacquainted with geometry." The analytic or modern method is, as to its form, characterized by the application of the processes of algebra and the calculus to the discussion of the relations of space. But its true nature consists in its generality. The an- cient geometry was essentially special. Thus the study of one curve was of little or no advantage in the study of another, except in so far as it had trained and strengthened the mental powers. The problem to draw a tan- gent to any point of a curve affords striking example of the difference between the two methods. When the ancient geometer had discovered a method of drawing a tangent to any point of the circle or the ellipse, this did not aid him in drawing a tangent to the curves called the conchoid and the cis- soid. Whenever a new curve was discov- ered, the problem of drawing a tangent to it had to be solved anew, and independently of its solution in the case of any other curve. Modern geometry substitutes, in place of the consideration of the geometrical magnitudes- themselves, the consideration of equations rep- resenting them according to a general system ; and after the discovery of the differential cal- culus the problem above mentioned was solved with the greatest ease and simplicity by a formula applicable to every known curve and to every curve that may hereafter be discov- ered or invented. (See ANALYTICAL GEOME- TRY.) Considered as a method of arriving at results, the modern is infinitely superior to the ancient ; considered as a means of mental dis- cipline, its superiority is disputable. The his- tory of geometry may be conveniently divided into five periods. The first extends from the origin of the science to about A. D. 550, fol- lowed by a period of about 1,000 years du- ring which it made no advance, and in Europe was enshrouded in the darkness of the mid- dle ages ; the second began about 1550, with the revival of the ancient geometry ; the third in the first half of the 17th century, with the invention by Descartes of analytical or mod- ern geometry; the fourth in 1684, with the invention of the differential calculus ; the fifth with the invention of descriptive geometry by Monge in 1795. The quaternions of Sir William Rowan Hamilton, the Ausdehnungs- leJire of Dr. Hermann Grassmann, and various other publications, indicate the dawn of a new period. Whether they are destined to remain merely monuments of the ingenuity and acute- ness of their authors, or are to become mighty instruments in the investigation of old and the discovery of new truths, it is perhaps im- possible to predict. According to a tradition handed down by the Greek historians of ge- ometry, the science took its rise among the Egyptians. The inundations of the Nile an- nually obliterated their landmarks, and efforts to restore them gave rise to geometry. From them, about 600 B. 0., Thales of Miletus, one of the " seven wise men " of Greece, is said to have derived a knowledge of the elements of geometry, and to have introduced it into Greece. Pythagoras is also said to have de- rived his first notions of geometry from the same source, and to him is ascribed the dis-
